will this new 7-speed chain work on a 10-speed cassette?
That's based on my experience with a nine-speed chain shipped by mistake on a 10-speed drive train. That didn't work well - shifting was horrible at best and the chain wouldn't stay seated on some gears. If a nine-speed chain does that, a 7-speed chain will be unusable.
If it even fits through the rear derailleur...
Also are QuickLinks compatible between 7-speed and 10-speed?
If you put a 7-speed quick link on a 10-speed chain, the wider link will interfere with shifting, and if you're lucky that's all it will do. If you just have standard luck, the wider link will catch on something like your rear derailleur cage, rip your rear derailleur right off, toss it into the rear wheel spokes, and break about 1/3 of those spokes, bend/kink about another 1/3 of them, and put gouges into the remaining 1/3 of your rear spokes so if you reuse them they'll fail quickly at the newly-minted stress risers.
And then you'll crash.
A 10-speed quick link won't fit on a 7-speed chain.