How many transistors can be connected to Arduino pins?
The TIP122 is a 100V, 5A, Hfe at least 1000, darlington NPN transistor. With the base resistor of 2k2, that is about 1 or 2 mA per pin. You don't have to worry about a few mA.

To answer your question: "How many transistor can be connected to arduino pins?", that will be about 100.

Why do you use TIP122 ? They have a voltage drop. Logic level mosfets have a very little voltage drop and thus they stay a lot cooler.
Mosfets leak at a higher voltage. If you need to operate near 100V, there are IGBT transistors for that.

Make sure that the high current through the TIP122 at the emitter does not disturb the GND of the ATmega chip. You have to design your circuit board to avoid such troubles.