Does any country have free college & open admissions?
Here is an overview of the situation in Germany, where there are no tuition fees for Bachelor and Master programs at public universities (though student union fees and public transport fees totaling 60-130 EUR per semester still apply).
For Bachelor degrees, the general requirements for admission to any German university are (1) proof of knowledge of the language of the study program (German or English) and (2) university entrance qualification (Hochschulzugangsberechtigung).
"University entrance qualification" means proof that you are eligible to attend a university. In Germany, this is obtained by completing the Abitur examination. In the German education system, at the age of 10, students are already separated into different types of schools based on their performance, some of which have direct access to Abitur (and therefore to university), and others which have practically no access to university. About 50% of German students complete Abitur and have directly access to university.
For international students, this is the corresponding high school leaving certificate. Depending on the country, your school leaving certificate might directly qualify you to study in a German university, or you might be required to do a two-semester preparatory course (Studienkolleg) before being allowed into German university . In this database (only in German), you can search for your country and school leaving certificate to see how it is in your case.
For most Bachelor degrees, admission is unrestricted (Zulassungsfrei), which means that any student that satisfies the above general requirements will be admitted, without going through a field-specific aptitude test. Switching programs after the first year is not difficult, but there is a somewhat strict requirement to pass the first-year courses in a limited amount of time, so the idea is to let students "test" their preferred program and switch to another one after the first year if they are not comfortable with it.
Note that not all Bachelor programs have unrestricted admissions: in some fields such as Medicine there is a limited number of admissions and students will be chosen according to their grades in previous stages. This is called Numerus Clausus (NC).
On the other hand, most Master programs do have an aptitude test and students are admitted based on qualification in their field (in addition to the general requirement of having a Bachelor's degree). At this stage, students are supposed to already know what they want to study. I would expect the level of "strictness" of the aptitude test to be in some way related to supply and demand (disclaimer: personal opinion here), so the requirements do not necessarily have to be stricter than in US universities. The offer of Programs taught in English (which are mostly only Master programs) is more limited than for German programs, so the aptitude test for English programs might also be stricter.
As pointed out in the comments, many Master programs are the result of the splitting of old 5-year programs (Diplom) into a 3-year Bachelor and a 2-year Master. Students doing the Bachelor program that apply to continue with the corresponding Master program at the same university usually have very easy admission into the Master program.
In summary, the requirements for entrance to German university in general are somewhat strict, but once you are in, there are in most cases few additional requirements for choosing a specific university or field.
Well, it depends how strict you are with the terms "open" and "free":
Open:
Most European countries I am familiar with (Austria, Germany, Sweden, Switzerland) are actually fairly open in terms of admission. There are always some basic requirements (e.g., candidates need to have a high school diploma or comparable), and sometimes there are entry examinations, but in general there is no US-style admissions system.
As an example, in Sweden university programs can define their own objective admission criteria (e.g., specific grades in specific high school subjects) and how many students they are willing to admit, but they cannot rank students themselves - they need to accept at least as many eligible students as advertised based on a ranking generated centrally (usually based on high school grades, as far as I know). Students will generally get admitted somewhere, but some of the best schools (e.g., KTH, Chalmers) you will need rather good grades to get in.
In Germany, the system is comparable to Sweden in the sense that your high school grades largely determine whether you can enroll in a specific program at a specific university through a system called Numerus Clausus.
In Austria, historically every eligible prospective student could enroll freely in any program of any university (there was no further admission system, aside from basic requirements such as having a high school degree or equivalent). This is still the case for many programs, but in some high-demand programs (e.g., medicine) they have now started to experiment with entry exams.
These systems are all quite different to US-style admissions systems. There are no application packages, interviews, or recommendation letters, and it's generally not the university that actually makes the decision (instead there is some broken but objective kind of ranking criterium, such as grades or test results). Further, these systems are often more about routing than they are about admission - for instance, in Computer Science in Austria, everybody who wants to study CS can do so in Austria, but not everybody will be able to study at TU Vienna (historically the most popular place to study CS in Austria). It's also important to understand that the difference in quality of the schools is actually not dramatic, so "getting into" TU Vienna is not a huge job market advantage - people mostly prefer studying in some cities over others.
In Switzerland, as far as I know admission is still completely open - every eligible Swiss citizen can choose to enroll at ETH or EPFL (international top universities). The "selection" here mostly happens at the first semesters of study - for ETH, I know that many, many students will give it a shot and drop out within the first weeks, after realizing that they will likely not be able to keep up with the highly intense workload and intellectual challenge.
Free:
Education in all these European countries varies between free and (relatively) cheap. Taking Austria as an example again, studying was historically indeed free at all public universities (which were basically all of them - private universities are a new trend around here, and not one that has really gained any traction). Since around 2000, we have been flip-flopping back and forth between "free" and a study fee of around 400 USD per semester (depending on which political party was governing the country). Tuition for non-EU citizens is about twice as high (so about 1600 USD per year of study).
In Sweden (and, I think, in Switzerland and Germany) there is no fee for eligible citizens (EU citizens in the case of Sweden and Germany, Swiss citizens in Switzerland). At least in Sweden and Switzerland, tuition for citizens of other countries is at least in the same ballpark as in the US (still a bit cheaper, but far from "free").
However, I assume that placements are competitive and have strict entrance and performance requirements.
To summarize, this assumption does not really work out. Systems differ, but in general universities are fairly open for eligible students (although entry tests and grade-based systems are increasingly seeing use).
So how do universities cope? Mainly by scaling up. Classrooms in Europe can be large - classes with 500 or 1000 students are not a rare sight in popular programs in Austria, Switzerland, or Germany (but not in Sweden, as programs can limit how many yearly students they accept - often in the range of about 100, give or take). It is often accepted that many students will fail and drop out - when I studied CS in Austria, it was not uncommon to partake in early exams with fail rates considerably above 50%, and about half of my classmates switched programs / universities during the first year of studying.
I have taught at both at two Swedish universities and one American university (Ivy League). The main difference is really the philosophy - in the US, it is difficult to get in, but once admitted, it is not that hard to stay with passing grades.
In Sweden, getting admitted is relatively easy (which implies a much more diverse body of students), but the difficult part is actually passing the classes. In mathematics, only about 40%-70% pass the final exam (passing grade is from A-E). Not having to pay tuition might explain why students are not as motivated as in the US, but studying without eventually getting a diploma is a waste of time that could have been spent on working.
I still prefer the Swedish system - it allows for people with diverse background to have the opportunity to get a diploma. Only very few programs have a high competition for the spots, so higher education is available in principle for almost everyone. But to stress - passing the classes is the main challenge and I would say that the quality of the education is at least on par with the United States.
EDIT: There are several ways to get a spot at a Swedish university.
High-school diploma that fulfills some basic requirements depending on program (these requirements can be fulfilled later, for example in math, many universities offer preparatory courses as a complement to high-school diploma).
University entrance exam Anyone can take this exam, any number of times (offered twice a year), each time at the cost of about $50.
Work experience Military service, having a company, or general work experience might give enough support university admittance.
I have taught classes with people between 18 to 60 years old -- as mentioned, it is quite diverse. It is also quite common to work part-time in parallel with studies. Also, students get a scholarship for university studies (unless their income from working is already over a certain limit).
I noticed that during 2009-2011, the classes were larger. It is most likely due to the 2008 economic crisis, and thus fewer work opportunities. Universities hence act as a sort of 'buffer' to keep people busy -- it is better (IMHO) if the tax-payer money to goes towards university tuition, rather than unemployment benefits.
On grad-school: Another big difference is that graduate studies (getting a PhD), is a proper job in Sweden (with unemployment benefits, parental leave, sick leave, retirement savings, etc) and the salary is something one can live comfortable off. Also, it is much less that a PhD student prematurely stop the studies, compared to the US system, where the qualifying exams are expected to get rid of say 50% of the PhD students.
In the US, it seems very difficult to handle a pregnancy during the PhD studies.