Though it isnt nevessary, the more you use it the more fluent the action becomes! Lever actions will eventually loosen up with use. So keep slinging lead and you'll be much more than just trigger happy ;)

Lever actions are like wine, the older the better, as long as you use them lots and clean/oil them correctly.
The more you use them the smoother they become. Just make sure you give it the right care.
Shoot lots and enjoy!!!

Nope. Just always clean and lube a new firearm. Lever action rifles are made to go right out the box.

When shooting a new lever action I always give it a good inspection. It's usually the action that needs some breaking in. Like Larry said about the milling marks and rough surfaces they need to be smoothed out. I've never had to have any work done they usually come out by working the action or taking it to the range and shooting it. I have had to use a little graphite oil on one of my new rifles but I recommend cleaning it well after it's smoothed out to rid the access graphite oil.

Terry W basically answered this but heres my take, firearm break-in goes as follows (assuming it is needed):
1. Barrel - Though this is contested it is often considered good practice to break in a barrel. However since lever action rifles tend to be for hunting only, and at short distances it does not make much difference. You would be hard pressed to find much difference between a lever action rifle that has been broken in and on that has not. You would probably be better off just working to keep your barrel in equilibrium (though this is contested as well).
2. Action - Whatever action it is, as Terry noted, there are often milling marks or rough surfaces (hence the reason people polish trigger surfaces etc). So smoothing here is really more of a smoothing out than a breaking-in.
3. Lastly there are other aspects that are probably more important. After working the action to smooth it, and so on i would just make sure it is correctly lubed and cleaned after use. These will probably do more for you in the long run.

No, not really as pertaining the barrel but spending some time working the action will definitely help sooth out the action. There are some milling marks on certain parts of both Marlin and Rossi. I have both and the Rossi definitely started to smooth out after working the action more.