I have a Moen single handle kitchen faucet that I cannot tighten adequately due to lack of room. I can get some turn on the lock nut with needle nose vice grips but not enough to completely hold it in place. Does anyone know if there is a special tool for this? Also should I be caulking the small gap between the faucet and sink bowl?
You could try one of these tools to see if it would get up in there to tighten the faucet anchor nut.
It would be used something like this:
I also wonder about whether you should have first removed the rusty old washer (A) and the dried plumbers putty (B) before installing the new faucet.
First off it seems very likely that the replacement faucet would have come with a new washer. And secondly it would be much easier and secure to tighten the nut against a new washer surface.
Usually you put a bead of plumber's putty around the faucet base to seal where it meets the sink. When tightened the excess should squeeze out, which can be wiped away.
It looks like there are #8/32 or #10/32 tapped holes in the nut. I agree with Michael, put a new washer under the nut, then try a basin wrench or put a screw in these holes to gain more leverage. Alternatively, put screws in all three holes, and use those to tighten the whole thing with a screwdriver, instead of a wrench. A new washer is definitely a good idea if going this route.