I've skied at Mammoth for many years and hopefully can help.
Spring is generally pretty awesome, and this year looks to be at the high end of awesomeness given the current 8-12' base. We go every year for our spring break (although ironically, we are tentatively heading your way to Colorado for spring break this year).
Last year we were there in April for our spring break and actually had some powder days because a storm came through. However, you can end up with some slushiness in the late afternoon (if it's warm there are some spots that get slushy--we just stay high on the mountain until our last run and avoid this).
Answers to your specific questions:
1. The 5 of you should have absolutely no problem finding skiing for all abilities; my family is 6 of varying abilities (with several black diamond skiers) and we do 1 week trips there all the time and never get bored. There are so many different lines to try off of the top ridge (from the upper gondola and Chair 23) and also off of Chairs 5, 9 (Cloud 9 Express), 3 (Facelift Express), 23 and 25 (and more) that you never run out of new experiences. There is an unlimited variety of blues all across the mountain to be explored as well. So terrain-wise this should fit the bill.
2. For lodging, Mammoth has lots of condos and the commonly used differentiation is "slopeside/walk to lifts"; "in town on shuttle"; and "in town not on shuttle," with pricing that is generally proporionate to how near you are to the lifts. There are technically 4 base areas: Main Lodge (the only lodging there is Mammoth Mountain's lodge, so you pay resort-type prices); Canyon Lodge (there are a number of private condos close to this); Eagle Lodge (ditto) and The Village (which is on the edge of town but there's a gondola from here to Canyon Lodge) (the Westin and some other high-priced lodging is here, as well as some private condos). Note that "town" is several miles from the main lodge and one or two miles from Canyon and Eagle lodges, so most dining options and stores are not near the lodges (there are restaurants in and near the Village).
Lodging-wise, for my family for a spring trip, I like to stay in a condo in town (better pricing, near restaurants and stores, typically no weather issues (note that Mammoth does a great job of keeping its roads usable in storms, but chains are usually required)) and base our skiing out of Canyon Lodge. What is awesome about Canyon Lodge (which doesn't seem to exist at any other resorts that I've visited in the west) is that they've got pay ski lockers, and we just get 2 of these at the beginning of our trip and keep them all week and store all of our skis and boots in them each night so that we don't need to schlepp them to and from the mountain every day [ASIDE--many resorts have an overnight ski check for about $5 per pair of skis (and you've still got to carry your boots somewhere); you can see how this can add up for a family of 6; with the ski locker, you can (with efficient packing) store 3-4 pairs of skis, poles and boots in a single locker for (I think) $2 per night]. Unless there's a major storm, there's no problem driving to and from Canyon Lodge each day. As an alternative to this (if you want to be at a "walk to lifts" spot) I would suggest that you find a condo near Eagle Lodge or Canyon Lodge. There are several Mammoth-specific rental agencies that represent condo owners and you can plug in your date and other requirements (no. of bedrooms, preferred location) and get a list of options. Also, a number of Mammoth condo owners do direct rentals through VRBO and similar websites.
3. Getting there--this can be the problem, and this is what (in a sense) makes Mammoth so great for those of us who live in LA because it cuts down on the number of travellers from afar who come to Mammoth (no offense meant; but less travellers from afar=shorter lift lines). In winter the road from Reno is often closed, but by spring, absent a big storm, I would assume it is open. It's 300 miles from LA to Mammoth so that's a 5 hour drive (through some pretty spectacular high desert with great views of the eastern Sierra and the mountains that border the west side of Death Valley; there is no mountain driving on this route, although in really intense storms there can be snow). There is air service to Mammoth but I think it's pretty pricey (you should check, though). Mammoth is not really accessible from the SF Bay area during winter because to get through the Sierras you've got to go via Tahoe (so why not just ski there, and that's an awesome alternative) or on a very long southern route that takes you far out of your way.
Good luck. If you are actually going there and you want some specific condo recommendations feel free to PM me.