BB choice to control chainring position & Q factor

I’m in the early stages of my first fixie conversion, and am trying to figure out how I want my chainline positioned. I’m intending on using a flip-flop track hub and 3/32" chain/sprockets, along with one of the chainrings that the bike came with.

Now, it’s a road bike with 7 speed 105 components, so Sheldon says it should have a 113-115mm bottom bracket length. What I’d like to confirm is, can I replace the BB with one of shorter length in order to move the chainrings inwards (bringing the outer ring inline with the rear sprocket) and reduce the Q-factor (giving greater cornering clearance)? And will any old square taper BB do, or are there other issues with fit between different square taper BB and crank models?

Thanks.

Yes, you can use a shorter bottom bracket.
Most square taper bottom brackets will do, but the taper will need to be JIS not ISO, hence shimano, sugino, tange = yes but campagnolo and miche = no.

Using a 113mm bottom bracket and putting the ring on the outside will yield a chainline of 46mm.
Most track hubs should have a chainline of 42mm.
To get a 42mm chainline you would need to use a 105mm bottom bracket.
Sugino/Tange make a 103mm BB, Shimano make a 107mm BB, any of which would get your chainline within 1mm. You could also use the shimano BB and space out the rear cog 1mm using a spacer. Of course not all cogs and hubs are the same and can affect chainline by a few tenths of millimetres anyway.

One possible problem you can run into is that the cranks may come in far enough that the inner tabs on the spider touch the rear stays, but in this case you can file them off
I used a 107mm BB instead of the 113mm BB when I installed my shimano DA 7400 cranks and had plenty of room.

Thanks, that’s perfect. There’s (just) enough clearance with the current BB to run the big chainring on the inside of the spider, so I should have plenty of space with no chainring on the inside.