The M-1 visa is for training, and is specific to the flight school you're doing the training at. Training would be regarded as any flight training towards a certificate/qualification, but purely renting an airplane does not require a visa, and this is why many people doing only time-building here in the states do so on a visa waiver, or tourist visa. There may be "gray areas" regarding certain endorsements; ie, tailwheel, complex etc as some training would be required in order to get signed off... but this doesn't necessarily come under the same category as training for a certificate, so this also should be allowed, and I've seen it done so I'm guessing it's "kosher".
So if you went to another flight school, did a rental checkout etc, then rented their airplane, it wouldn't be classed as flight training towards a certificate and should be allowed regardless of whether you have an M-1 visa or not. So yes, you can rent elsewhere as a private pilot (with instrument) and just be subject to that flightschool's rental requirements.
If I remember, TSA clearance is only required for Instrument Training, Multi-Engine, and/or training in airplanes weighing more than 12,500lbs. I believe that this is also school specific, ie you're cleared to do the Instrument/Multi at one flight school, so couldn't go do it at another school when you're current school authorized it and you have the relevant clearance.
As far as proficiency, take a look at the regs... It's there for all to see, and you should really know (or be able to find) the answer to your second question. (90 day rule to carry passengers, with 3 full-stop tkoff/ldgs & instrument currency for IFR) Most flight schools would require a check out, or may have their own limits imposed, such as 60 days (21 days for tailwheel etc) for currency to rent their airplanes. Question "2" purely depends on the school, it's up to them as it's their airplanes... ask them