Agree with what Thomas is saying. A few specific points on your questions:
1. TMax developer was formulated to give slightly more speed out of the TMax films. However you do not need to use TMax developer at all. There is no such thing as optimum results. In fact the benchmark developer for Kodak's testing of the TMax films was plain old D76 1+1 and the TMax films perform extremely well in D76 (or Ilford ID11). You can use virtually any general purpose developer with these films as long as you figure out the right development times, agitation schemes etc.
2. If you want to dilute TMax developer further than 1+4, expect a slight increase in grain. You'll have to experiment with longer development times to get the same speed and contrast as if you used it 1+4. Higher dilutions (1+9, 1+15 etc) are most often used in combination with reduced agitation for contrast reduction. For standard work I would recommend the 1+4 dilution with this developer. My advice is always to only dilute the developer for specific purposes and characteristics, not to save money. If you want to save money, buy a cheaper developer.