1-Veronica can stick with you for a long time of exploring/side quests regardless of what faction you're playing with, but there is a certain point in the game where a main-plot decision you must make will cause her to leave you. I think it applies to all but one of the main-plot faction paths but not 100% sure on that. With the NCR there is supposedly a complex way to avoid it if you know how, but I haven't tried it myself.
2-When Boone "warns" you, it's just a game warning to let players know if you do want to side w/Legion or do things peacefully, you can't do it with him along. He only does it the first time you approach certain towns/areas. They're not important otherwise.
3-Companions have a "companion quest" that is accessed by doing certain things/events in the game while they are with you. For Boone, this means you have to gain 'rep/history' points with him by doing certain quests in a way he likes. Most of the time you'll know it is or was an opportunity for Boone rep because he'll make a brief (non-menued) comment either as you get close to the quest area or after you've finished it to his liking. But some of them give no indication so.... :/ There's quite a few of them however, so usually not a problem to get what you need. But it can get confusing/frustrating 1st time, when there's so many side quests, heheh.
At any rate, every time you get some points, new convo options pop up. He won't let you know he has new convos, you have to check via the Wheel. With enough points, he'll eventually give you his quest. And yes, it does involve killing some Legion.

Oh...and you have to have talked to Manny (the other sniper in Novac) about Bitter Springs, or the convo's won't show up in Boone's wheel. I don't know if this is a bug or intentional.