Poor girl looks like she is suffering from a serious potassium deficiency. Needs bacon ASAP (just joking - but c'mon giver her some bacon)

Are they easy to train?

Yes and no. They can be headstrong but they absolutely thrive with regular consistent training. A puppy class is probably going to be must and after that I would recommend having a regular training schedule

Any health issues that are common with the breed?

Heart issues can be quite common but are becoming less so with better breeding programs. They are also prone to cancers. Nothing to stress about though - just make sure you stay on top of your vet checks.

Should we dock her tail?

Up to you. It's not an option where I live so not something I can really comment on. Their tails can be weaponised when excited though.

Socialisation will be vital. Make sure you get socialisation in both with and without your other two. Boxers can be quite protective of other family dogs so you need to make sure they know how to socialise with other dogs on their own appropriately, not just as part of the "pack"

Boxers are also infamous jumpers. With a beagle (and a shar pei) already in the mix I hope you have good fences.