It's not that close. The moon is 384,400km away, and JWST will be 1.5 million km away (i.e., around 4x as far as the moon). It would be a pretty serious undertaking to get to it and get back.
Distance isn’t the best proxy for difficulty of a space mission. The delta-v to get to low lunar orbit is higher than the delta-v to get to L2. The delta-v required for lunar landing is considerably higher. And all of these manoeuvres are much less than the delta-v required to get to LEO.
That depends. If we're talking a manned service mission, then distance is a factor as it would mean a several month trip using current vehicles.
That being said, and I haven't done the math, but I think a Falcon Heavy and Dragon could do it at least in terms of delta V; although there are numerous problems with that like the fact that the dragon doesn't have a proper airlock for starters.
