I've been in early June three times now. Here are my experiences--
-MK: Fastpasses for Splash and Big Thunder usually aren't too hard to get in the afternoon. Return times are usually further away for Splash. With Big Thunder, it seems like the return time usually isn't more than 90-120 minutes away. Space Mountain FPs usually go a lot quicker-- if there are any left buy early afternoon, they're usually for much later in the day. The hardest FPs to get in the afternoon, though, are at Peter Pan's Flight-- they seem to run out quickly, and the standby line is usually really long in the afternoon.
-Epcot: The hardest FPs to get are for Soarin', since it is so popular and slow loading. If there are any FPs left an the early afternoon, they're usually for late in the day (8:00-9:00). Test Track used to be like that, but it's not quite as bad now since Soarin' has diverted some traffic away (plus you can use the singles line there if neccesary). I wouldn't worry about Mission: Space at all, as the wait usually isn't more than 10-15 minutes most of the time anyway (and less than that in the singles line), I guess due to people being scared off (which is good for me, it's my favorite attraction at Epcot!)
-MGM: The only hard-to-get FPs are usually at Rock n' Roller Coaster, but in early June you should be OK. Tower of Terror is pretty easy to get Fastpasses for, as are all the other MGM attractions.
-AK: I haven't had much problem getting FPs for anything here. Never experienced Everest yet, so I don't know how that goes, but I'm assuming they'll go quicker than at the other rides here.
Hope this helps, and hope you have a great trip! We'd normally be joining you then, but we had to push our trip to July this year. First week of June crowds are not as bad as mid-July.