The question is really: is the offer they are giving you fair, and is it more or less than you had expected? There is nothing wrong with squeezing them for more if you can get it, but in general that is something you do once you have established yourself in an industry, not when you are starting out. If you already feel this is a good contract, then you are just being greedy (not that there's anything necessarily wrong with that) by asking for more, and this may sour them on bringing you on board at all.
If you think their offer is below what you are worth, however, I would ask for more AND be prepared to turn them down if they don't budge. It doesn't sound like there is any big hurry for you to get a job lined up, so don't be pressured into taking the first thing that comes along if you don't have to. Just be sure that if you say no, you give them the conditions on which you will accept, because you never know if upper management will intervene at that point and cave to you - I've had it happen for me, so it is possible.
As far as opportunities within the organization go as an incentive, I would say that in the majority of cases you progress further by switching organizations than you do by sticking with one, as loyalty is rarely rewarded, despite what promises the recruiters may give (who are almost never the same people who actually do the promoting). One way I determine if such promises are smoke or not, is to ask the person interviewing me how long they have been working with the organization - if they are middle or upper management and have been there less than a couple years, you can almost guarantee that their rhetoric about you have the chance to "go places" has little credibility.
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