You can put the valve anywhere. You could put the spout on the ledge of the tub, or on the wall. There are some that have a long reach and some that come out as a waterfall. For filling a large tub, I'd want at least 3/4" pex or maybe even 1", then reduce to a (preferably) 3/4" valve. If the supply lines into the house aren't at least 3/4" copper (and preferably more that goes all the way to and through the WH), you'll be dissapointed in the fill time. While 1/2" pex to the tub would work, the gpm would be moderately low. The ID of pex is smaller than copper, so 1/2" pex can flow less than 1/2" copper. Not a problem for most things, but a multi-head shower, large tub, outside hoses, etc., anywhere where you want max flow, you may be dissapointed with 1/2". 3/8 to a vanity sink is probably fine, since those valves are flow-restricted anyways, but not a tub filler.
Yes, larger pipes will take longer to purge of cooled off water, but you also have the option of recirculation, and then the delay is non-existent (or at least small).