I can get behind here rather fast due to being in the USA...so sorry if its been asked and answered already.

Watching a repeat of Flesh and Stone. the end River mentioned her handcuffs and asks the Doctor why it must always end this way for them.

#10 was in handcuffs at when River dies...but she ain't gonna know that yet.

does this mean that there's a running gag-ish type them of them always parting with one of them in handcuffs? lol

Yes, it's to be assumed so. Now how could their next encounter end in handcuffs? Hmm, maybe with the killing of the greatest man she ever knew

Oh, I can think of other reasons for handcuffs ...

I assumed this was the implied meaning of the line in question...

You are very likely correct! It's late!