mmo: well that would be a completely different game in and of it's self. (mahaha completely side stepped that one didn't I!) but no i think it should work in this way.
take a Q from Guild wars. the towns/outposts will be like an mmo. while outside is only you(and who ever is in your group) that why(i think) every one is happy. the single player, players(eh how that term is right) get their single player game while the mmo fans get theirs. if you just want to go off by your self fine do that. if you want to team up then do that as well.
taking another Q from a game Diablo 1 and 2(3 mite have this as well) depending on how many players are in your group the enemies in the world will be tougher (i'm going to say 10% for each player other then your self.)
multi-player: now for regards of multi-player if there ever is one for the normal fallout game(fallout 4!) here is my thoughts.
multi-player battles: i think it would be cool not to have the multi-player option on the start menu/main menu but have it so you have to go to an area in the game it self to connect for multi-player battles. each area(map) has to be entered though a loading screen, then like any other multi-player battle game you have your team screen where you can join the team you want.
each area(map) has its own play style. one is meant for CTF(capture the flag) while the other one is meant for BTB(big team battle) and and is meant for DM(death match. free for all, every one vs. every one.)
entering the multi-player battle maps, you cannot loot other players bodies(and they you) how ever for winning(or being on the winning team) you get items or caps(depending on how good you did.) and also some amount of xp. aslo when in the maps your equipment does not degrade.
co-op: co-op will work like any other game in that has co-op. you have a friend at your house/or on live and you invite them over to your game.
taking a Q from fable 2 the player that comes in on your game has a pre-made character in the game. if they have their own fallout character then those stats on that character are transphered over to the pre-made character. and just like just like fable 2 you have a safety mode so the guest can't kill friendly NPCs(unless you start first) the guest(s) have to stay within 100ft from the host or a mini count down will start if the time runs out then they are booted out of the game.
how many players?: i'm thinking 2-4 players is that seems now that is what games are going for is 4 players.
again like in the mmo mode the enemies will be increased in toughness by 10% per player(in co-op mode)