1) Mog station items are tied to the character (or all characters). If you've left them in your mog mail, that's where they'll be pre and post transfer.
2) It depends. If you're moving from a Congest server or to a Preferred server, the limits are changed. Otherwise, there are still caps. Up to date details should be on the transfer service section.
3) Yes, but to get them back on the markets, you'll need to manually pull them up again. I'd suggest taking items off for sale before you do your last logout before transferring. The same goes for leaving your Free Company and relinquishing any property, if applicable, but there are exceptions.
Do check
Lodestone Page on Transfers for the most up to date information!