Can I connect my laptop that runs on 18.5V directly to a 12V battery?

There is no way to know from the outside how the laptop will react to 12 v, but probably not well. It may also not react well to having the battery completely removed. It could possibly be integral to the internal regulation.

The safe thing to do is to get a "car charger" for your laptop. These are intended to work from 12 V car power and supply whatever power your laptop needs.

Can I connect my laptop that runs on 18.5 V directly to a 12 V battery or 24 V (2 batteries)? That's 18.5 V on the laptop's DC plug. Laptops have voltage regulator circuits on the mainboard so they can accept slight differences in input voltage, but supplying 6.5 V less or 5.5 V more to a 18.5 V DC plug is very probably too much of a difference. For example, I have a ThinkPad X201 notebook (20 V DC nominal) that I can operate reliably with 18.8 V or 20.4 V DC, but supplying only 16.4 V DC will cause it to not even start. Other computers may react differently, and it might not be safe in all cases.

Since the internal battery outputs 10.8V, should not the laptop be able to run on that low voltage from the [DC plug] power source as well? No, not necessarily, since there are voltage converter circuits on the mainboard so that the supply voltages expected from battery and DC plug can indeed differ. Again for my ThinkPad X201: it has a 14.8 V battery, but would not even start up when supplying that voltage to the DC plug. Now you could try of course supplying the battery voltage (or something in the operating voltage range of the battery which is 9 V - 12.6 V for a 3s2p battery as you have) to the battery terminals. But again there's a problem: a typical battery connector has five or more pins. Plus and minus are usually at the very left and right, but there are data pins as well that let the laptop communicate with the battery. In my ThinkPad X201, just supplying battery voltage to plus and minus did not let the computer start up, since it did not recognize a "proper" battery via the data channel. It did not damage anything either, but it did not work. I even tried connecting the data channels from the dead battery in addition, but that did not work either …

Proposed solution. As proposed already, a DC-DC converter that supplies your 18.5 V DC (or 18 V or 19 V) from 12 V DC (one battery) or 24 V DC (two batteries) is the clean solution to go for. Their typical efficiencies are 85-90%, and they are more efficient with less voltage difference to cover. Since both 12 V → 18.5 V and 24 V → 18.5 V has similar differences, I recommend using a 12 V DC adapter since it requires only one battery for a start, and since DC adapters for notebooks with 12 V input (as used in cars) are more readily available than with 24 V input (only used in trucks). There are cheap adjustable ones available, no need to buy the expensive official "12 V car adapter" made by your laptop manufacturer. And before you wonder about the 10-15% power loss in the adapter, there are many ways to save more power by using the computer in an energy-saving way (display brightness, monitoring for runaway processes, frequent standby, auto-power-off for the monitor etc.).