Any film can be developed by any developer, although you may get differences in results (stuff like sharpness, grain, accutance, contrast - I'm not familiar with all these). And yes, you can do it at any (reasonable) temperature. The other factors would be dilution and timing.
In order to simplify matters, it's best to keep two out of the three variables fixed, and change the third variable according to experience. In my case, I usually fix the dilution (1:100) and the temperature (room temperature, which is 27 deg in my aircon room).
The only thing that changes would then be the timing, for different ISO.
Agreed that TriX is more forgiving than Tmax. Have not tried Ilford films, so cannot comment. I tend to stick to proven formulas, which is 20 min for TriX 400 exposed at ISO 1600, and about 10 min for Trix 400 at ISO 400.
Some would question why I choose such a high dilution. Firstly it uses less developer concentrate (I'm referring to HC-110), and secondly, the timing is less critical. A difference of 1 min is a smaller fraction of 20 min than of 5 min. The higher temperature (27 deg vs 20 deg) somewhat compensates for the lower dilution, so you don't have to soak the films for too long (eg 45 min). The only drawback would be exhaustion of developer, which I suspect happens beyond 20 min.
There is one more variable I have not mentioned, and that is agitation. Since I stick to 10 sec agitation right at the beginning, that is one less variable to worry about.
I don't worry about the other film qualities (sharpness, accutance, contrast) because ultimately I scan my films and post-process in photoshop before printing, so I can adjust these to what I want. Why then do I shoot film? Good question.
