Yes Zoe, absolutely there is. One would need to protect the lysine nitrogen separately, and ideally be able to deprotect the lysine nitrogen under conditions that leave other nitrogens untouched. Functional groups that can be removed without affecting other existing protecting groups are said to be “orthogonal”, using an analogy from geometry. This is why such a large number of protecting groups have been developed.
For example one could use a benzyl protecting group (PhCH2) on the lysine nitrogen and remove it with Pd-C / H2 without affecting any Boc or FMOC protecting groups. One could deprotect FMOC with base (piperidine) without affecting the other two, or remove BOC with strong acid (TFA).
Does that answer your question? James

The usual approach is to protect the nitrogen(s) before you bring them together in the same molecule. If you’re in a situation where you have two chemically similar amines and have to selectively protect one, that’s a planning problem. There are a lot of variables so it depends on the particulars of your molecule, but there’s almost always a way to do it!
Whether you use Boc or CBz depends on the other functional groups in your molecule. The acid required for Boc deprotection can sometimes affect sensitive functional groups in your molecule; CBz deprotection is much milder. It would be easier to give you a more specific answer if you showed me a specific problem.

I don’t know. I’m unaware of any schemes for deprotecting the glutamine sidechain in vivo.

Other acids work too, such as bubbling HCl gas through the solution. However using neat (i.e. undiluted) TFA is just such a clean reaction that it’s typically the first choice. You can literally just dissolve your reactant in dicholoromethane in a stirred, open flask, get out a pipet, drop neat TFA into solution and wait for the bubbling to stop (the byproduct is isobutylene, aka 2-methyl-propene) and the reaction is done. It’s extraordinarily easy to do.