I don't have the rules in front of me, but the way I've been playing and have understood the rules is as follows:
1. Can you place your new station (the station that you deliver the white cubes to) before you place your string for the turn?
Yes, I believe so. I think the only restriction is that you can't move cubes or upgrade locomotive before placing string for the turn, but you can place new station, draw long distance tiles, add new cubes from the bag, or discard cubes to gain AP.
2. If you draw cubes for an AP, can they be placed on an unconnected Outskirts station?
Interesting, I haven't had this come up, and would want to check the wording of the rule, but my recollection is that you can freely add the two cubes to any tile (as long as they go together onto the same tile), so I would say yes, you can place on an unconnected Outskirts.
3. Can a white cube for a +5 point long route delivery begin in the first station listed on the card, or must it pass throught the first station, to be a legal delivery?
No, I believe it must pass through the first station, not simply start in the first station. So for 2 AP you could take it out and back into that first station to satisfy the first stop on the long distance tile, but I don't believe you can just take it out and continue on your way if you want that first tile to count.