2. There is a charge for valet parking at other WDW resorts, but you can leave your car as long as you want (i.e. not limited to the three hours).
3. Once you've paid for valet parking at one WDW resort, you can valet park for free (with tip, of course) at the other WDW resorts that same day.
We sometimes have a meal at a resort and would like to keep our car parked at the resort and use WDW transport to an associated park, but don't want to do so illegally.
On a couple of occasions when I wasn't staying on site, I just showed my DVC Membership card and parked no problem. Maybe I wasn't entitled to, but there was no problem.
Like a number of the discounts, valet parking is not tied to staying on points or even staying on property at WDW. You need either your card OR your room key that says member but not both.
You are also correct that if you park at a non-DVC resort and pay for valet parking that you will be able to valet park elsewhere for that day with no charge (except tip). Generally, I don't think there is a time limit on valet parking but during busy times they may put a restriction on the parking at some resorts (such as the Contemporary).