Usually they've checked out my website or my one-sheet when they're deciding if they want to take the meeting or not, so to be more clear -- I don't present my website. I go in by myself or with my producing or writing partner or director.
As far as budget goes -- if company I really want to work with wants to do my project, I'm gonna let them lead on the budget (and not propose one myself) -- at this point anyway. Perhaps that will change. If it's a production company I know has a lot of money tho, I'd be honest about what I'd need to keep the production value they're looking for.
Also, my personal opinion is don't bring in papers (ie scripts) to leave with them (you can bring your one-sheet yes, if that feels good). I'd tell them in person what your idea is (let them use their imaginations about what it could be -- stay flexible) and then they can request the script (which you can change if you feel it can meet your combined vision better after hearing what they're looking for) or you can follow up with a video example of your work.