How to ask for a letter of recommendation on a tight deadline, from professors I haven't spoken to in years?
This is totally fine, don't worry about it. I wouldn't "apologize for your presumptuous e-mail". You might apologize for the fact that you are asking them to write on short notice; I'd simply explain that you had solicited references from people who had interacted with you more recently, and schools to which you had applied had requested a reference from your undergraduate institution. 

Also I think it's fine to ask by e-mail instead of in person.

I'm unclear about what your relationship is to your recommenders -- you say that you finished a bachelor's degree earlier; are you a student or an instructor at this community college? In any case, you want to get letters from the people who can attest to your ability to do demanding academic work in your chosen discipline. It sounds like you might have chosen your recommenders poorly (i.e., you have chosen people who will say in generic terms that you are a "good person and a hard worker", or perhaps you taught you easier classes than you took as an undergrad). I speculate that your file is very positive but lacks credible evidence that you will succeed in a graduate program. If this is the case, then the school that contacted you is doing you a big favor (they could have simply rejected you).

You should ask someone who, in addition to thinking highly of you, will know what it takes to succeed in a Ph.D. program and will be able to communicate this in his or her letter.