Dollar amount of NSF grant How do you decide how much money to request?  Must you have a detailed list of costs?  How much money can be taken by the PI, for instance, for summer pay?  I really don't understand how the numbers are reached; they seem totally arbitrary to me.
For senior personnel (like the principal investigators) the NSF limit salary to 2 months salary for each person per year, which is held across all NSF-funded grants (so if you are listed on 100 NSF grants you still can only get 2 months per year for yourself in total).

As to the budget questions - yes, you'll need a budget, plenty of supporting detail and reasoning, and yes there are lots of rules about everything from computer equipment to salary for clerical workers. Most common questions can be answered by reviewing the NSF Proposal &amp; Award Policies &amp; Procedures Guide. There is no single rule of thumb on how much to ask for, as the range of awards varies quite widely (and for the NSF is usually stated in Dear Colleague letters or public grant invitations on specific topics), but in general if you don't have a really good idea of how much things might cost to go into the budget for what to ask for, the panel will probably quickly notice you haven't thought things out in enough detail and reject it easily in an early round. You can be reasonably assured that someone - and probably many people - will look at each line item in a submitted budget, so they will need to be sensible, clearly defined, and obviously necessary or highly desirable for ensuring the success of the project. If the budget given does indeed seem arbitrary to someone who has looked at hundreds of grant requests in the last month (or week), that will not bode well for the likelihood of success.

If you are affiliated with an institution that applies for NSF funding in general, there is probably a dedicated staff member (or a whole department) dedicated to supporting grant requests, so you should also definitely reach out to them to get information on what sort of support is available - which is much easier than trying to figure out everything on your own. They may have templates, workshops, guides, internal reviews, etc. - or at least, some institutions have all that. Check to see before doing it all the hard way.