Get an old Tektronix from ebay. They are solid units that will do what you need, and won't cost a fortune.
No (AVS) and no (Sunny). Azimuth adjustment is not about output per channel. It is about crosstalk. Although you can in theory at least use a 'scope to do azimuth, it is more complex than you might think (first you normalize the output of one channel vs the other and then you look at the amount of the L channel signal that appears in the R channel, and vice-versa) and requires filters so you can look at one frequency at a time, and of course a proper test LP. Same goes for using a voltmeter. That's why they get the big bucks for the Fozgometer or the Feickert device (which I like better in theory). Re the FFT business: I don't know much about that, but I do know that there are now applications that you can download onto your laptop that turn it into a frequency analyzer for very little money compared to the cost of a modern digital storage 'scope with FFT. (Actually, the Feickert program does that too.)
Re the PC based Oscilloscopes vs stand alone units - if you check ebay there are some used DSO's selling in the range of $250 to $500 which is about what a pc based product would cost, and the specs appear to be better on the stand alone units (especially sampling rate). I mentioned this if want a good general purpose scope - if you are just looking for azimuth measurements, a scope may not be the best approach.
A scope will let you check for phase differences between channels. Cross talk and phase differences should be at minimum at the same point, but only if the cartridge is perfect. Good ones are close, but there are duds out there. In any case the scope needs to be 2 channel with an X/Y input. Or you could buy the Feickert program that does both.