If I finish the campaign mode, will I get some benefit when I play on-line?

If you mean will you unlock badges, better weapons for multiplayer then no. They all have to be unlocked by playing multiplayer only.

The only benefit you'll get is from more experience with the game. Like you say, you'll be more familiar with the weapons and slightly more familiar with the maps (although overall layout will be different).

I'd definitely recommend playing through single player first just to get your bearings with the game, but be advised multiplayer is a whole different ball game to single player, good luck!

To me, the singleplayer is useful for multiplayer as it lets you gauge the feel of the game; what movement feels like, how guns behave, what control setup works best for you, how the game engine and graphics feel. It's good for stuff like recognising what bits of scenery are easy to mistake for players, what sort of height of wall must be mantled and what can just be jumped over and so on. If you like the fiction of the game world, it also lets you get some idea of who the teams/factions are.

Not really. As you said, you get to try a multitude of equipment, and if you're new to Call of Duty in general, it's sort of a tutorial on the controls and "combat" training.

However, apart from that, you don't unlock anything but achievements, and those don't help with multiplayer.

As has already been mentioned, the single player game doesn't have any direct influence on your multi-player experience. However, in MW3, the co-op missions can unlock titles -- one for reaching level 15 on all survival maps, and another for achieving 3 stars on all co-op missions. These are of course accessed via the single-player launcher and engine, and can be played single player if you are so inclined.