Would a professor write a bad letter of recommendation?

I would read these responses as: "I am happy to write you a letter of recommendation, I am busy, and just want know where to send my letter".

As a professor, I get literally hundreds of emails everyday, many of which, like yours, I want to reply to, but I don't have time to write elaborate or detailed responses. I read the responses you got as terse but professional.

If I am approached by a student I don't feel I can write a good recommendation letter for, I usually tell them this, and suggest they ask someone else. I think this is common practice.

This is the correct reply and just the way the exchange would have gone for one of my students. You should be pleased.

The professor will not need any information because the academic records and personal recollection provide everything that is necessary to put in the reference letter. If you had to provide information then the reference provided would be less worthy. Anyone can write from information provided, but to do it from personal experience will be so much more valuable.

The fact that reference will be sent direct to the colleges will give it so much more weight. A confidential reference from a professor that remembers teaching you is the best you could ever get.

Be more grateful and less paranoid.

Just to reiterate a point that is often misunderstood: a letter of recommendation that mostly refers to second-hand information is not positive. The best LORs refer to first-hand experience of the letter-writer with the student. What's the point of having the letter refer to things that are documented in transcripts and such? It adds nothing. Nothing. (Unless there's either positive or negative trouble/action that can be explained by adding information...)

So, when I write letters, the main point is to speak of my first-hand knowledge of the student's past accomplishments, my perception of their interest, motivation, and potential, based on my direct contact with them... and "looking them in the eye".

Email contact is more tenuous, sure, but, still, it has been known to be an adequate mode of communication "first-hand", not just a notifier of events elsewhere.

So, if your former teacher has a similar attitude as mine, then (s)he'd have no need at all for any further information... because his/her letter would, and should, be based on first-hand knowledge of you.