Building a simple mcu-driven array of LEDs
Yes
Yes. You can calculate the resistor by calculating base required current (you will need to know 'hfe' of your transistor). But you can just try. Working range is quite high, so if you will just use 500 Ohm for example, it should just work.

For example, if max total current is 100mA, voltage is 5V, and hfe is 150mA, you need 100mA / 150mA = 0.6mA base current, so resistor should be 5V / 0.0006 = 8.3kOhm or less.
There are lots of differences, but yes, it depends on how much current you will need in the worst case. If these LEDs are usual 10mA ones, almost any BJT will work so you can take what's cheaper or what you already have.


BTW. Don't see your shift register :-)