attach crossbar for ceiling suspended shelves using strong Simpson ties
That's going to depend on the fastener you use. For a vertically-suspended mount like this it's critical that you use a screw with enough thread area, or that you use through-bolts. I'd use 3/8" lag screws of 2-1/2" length or 5/16" machine bolts running all the way through the joist and nutted on top. Dividing your 400 lb. load in half gives us 200 lbs. per bracket, and 100 lbs. per screw. That's likely to hold well. Test safely first. Watch for deflection of the crossbar and deformation of the brackets. 
A 2x4 on edge will carry a substantial load, but I wouldn't hang that much weight further than about 16" from the brackets. If you see a lot of flex, move closer. 
See #1. Nails are almost never a good choice for load-bearing applications, and certainly not for direct suspension. You must use a threaded fastener of adequate diameter or a through-bolt with a suitable washer and nut.


I would consider using standard joist hangers and placing your crossbars between the joists, or simply lay your crossbar on top of the joists, if that suits your needs. Either is a much stronger method than vertical fasteners.