I just bought a A070 Shimano Sti 2x7 set for my bike which is 3x7. Could I still install and use all three chainrings? Or would I be limited to the two chain rings?
Yes - you have limited yourself to two front chainrings. Most likely you'd choose the big and middle, leaving you a useless granny ring. This could be removed for weight savings.
As a rider of a 3x7 who is also a poor climber, I love my triple and it lets be spin at 90 when a "modern" bike is mashing 60 RPM.
Consider returning the shifters and get what you really need, or go to a 2x7 and forgo steep climbs. Or you might be able to set the shifter on small and middle rings, then use something else to pull the inner wire more so the shifter mech pulls out to the biggest chainring. A wedge between frame and exposed downtube wire? Would look silly.