Upgrading stock front derailleur - Will Shimano Deore work with 3x7 gearing? I'm looking to upgrade the front derailleur from a stock DNP-brand (the ones every big-box BSO has) to a nicer Shimano. The goal is to reduce play and flex while switching, allowing a tighter adjustment and crisper shifting on the front chainrings. 

I think I want the Shimano Deore line, durable and stiff but nothing fancy, except that the full Deore groupset is based on a 10-speed rear cassette, while the bike currently has a 7-speed cassette and I'm not in much hurry to upgrade the full groupset (the only real problems I'm having with the bike are centered around this front mech; the other components, while by no means high-end, are getting their job done).

So, will a Deore FD-M610 work with the wider chain of a 3x7, 42-32-24 setup, or do I need to be looking elsewhere? I don't have exact specs on the chain at the moment, so I assume it would be the "standard" width for a 7/8-speed cassette which is in the 9/32" range.
A 10-speed FD will not work well. Mismatching FD "speeds" (inner cage width) with chain width usually gives poor results in either direction. In this case you would probably find it's impossible to get it to not rub somewhere. Just get a 7/8-speed Altus, FD-M313 for the current one. It will easily meet the goal you describe and play nice with the rest of the drivetrain without having to mess with anything else.