1) yes, it would
2) Yes, so long as you aren't using old legacy apps and small datasets, but you never mentioned the use!!
3) Cooler? Irrelevant. It either runs cool enough or doesn't. Does it run cool enough? Does your case have proper airflow? Most modern cases do have enough for the config you mentioned, unless you have extreme ambient temps or are trying to achive ultimate quietness, factors far too broad for a single topic.
4) Should you save up? Depends on how long you'd take to save up enough, what your ideal is of value, how long you'd use it till upgrading again, and primarily, what you do with the system, the most demanding apps, their versions, the data set sizes.
To put it another way, your best bet is listing what you feel are the most demanding things you do, AKA what now runs slowest that you want to improve, and listing more of your system specs, what you consider a reasonable budget range, and how long till you're upgrade again ideally.
Generally speaking, most who would upgrade to a Q9400 range CPU, are at the point where they'd benefit from more than 2GB memory, but not everyone... lots of people have a lot more memory than that mainly because memory dropped so much in price but at the same time that it dropped so much in price could also be an argument towards going ahead and increasing it unless your motherboard won't allow that... I don't know the specs on that board but suspect it does allow more than 2GB total.
I hesitate to advise saving up for an i7 though, not knowing how long you'd save I can generalize that since any price-point is a moving target in respect to the models or family you'd buy from, that ultimately the best value upgrade will be saving up approx $350 (ballpark) and at that point looking at what is available - unless you're willing to overclock in which case that 350 might drop closer to 270.