No you cannot complete MQ without being part of certain factions. You can join MM, but you have a choice to not do any their missions. Since this is G D thread I cannot give spesific hints, but you don't need to join any factions at all. Just take the Doggie for a walk and visit places. There are some locations that belongs to MQ and some of the NPC you may kill can mess up a little. There are locations that is closed, can only be accessible if you are a part of MQ. Doing faction missions and follow MQ gives you alot of xp, if you choose to be a lonely wanderer with "Meatloaf" you may not get much of xp. There are misc/independent quests to do, so go ahead
Cheers,
-Klevs
You can be a mercenary for all four factions that you want. Provided that you don't officially join by doing certain missions for the factions.
The thing is, you mentioned finishing the Main Quest. The Main Quest is primarily about what became of your kidnapped child, Shaun. ABSOLUTELY, you must join a faction to accomplish that.
Not necessarily to find him.
Nearly all of the quests in the game are built around the idea of relationships and picking sides. There's nothing stopping you from just wandering around with the dog and exploring but you're going to find very few quests that don't require you to get involved with people. The tale of an unattached drifter just isn't the story the game sets out to tell.