New to turntables - does this gear combo work?

Unfortunately, the answer is "maybe". The phono box is designed as what is called a line level device, to feed an input on an integrated amp or line stage pre-amp. The Pro-ject does not have an attenuator or volume control. The 9 volt specified output of the Project is quite a bit higher than a typical line level output of 2-3 volts, so it may be enough to drive the built-in amps of the A-2s. The Audioengine web site does not say what the input sensitivity of their amp is, so you will have to ask their tech support. If 9 v will drive it to full output, then you should be OK. Personally, given the generally non-technical nature of most tech support, I'd make sure that I had full return privileges before I went down that road. Your other alternative to the Project-Box is a full function pre-amp (phono stage plus line stage plus attenuator and input switcher) which will allow you to add a "gasp!!!" CD or tuner or computer line level source to the mix.

Assuming your TT comes with a cart., the system should work with no problems.

Swamp is correct, you should consider a real preamp for best sound ..do not over look this important stage it is the most profound effect on how well the system will sound besides speakers. A good sounding preamp is what your system really needs to bring out the dynamics and soundstage.

If the speakers have a volume control you should be ok!