and then delete it from my phone later, can I re-download it (for free of course) on my iPhone's iTunes store???? Or will it charge me again?
Yes, you can delete it from your iPhone (in the same way) and also download it again for free, either from your iPhone or from iTunes on your desktop iTunes.
Yes, you can, and no it won't charge you again. It will either say, "This item has already been purchased. Would you like to re-download", or if there has been an update since you deleted it, "This is a free update since you have already purchased this item".
I basically do it that way every time I buy a new app. I purchase it on my computer's iTunes, then download it again "for free" onto my Touch. I rarely actually sync the two.