Daughter should be possible, cousin would not.
Yes you can. Normally, I prefer K-1 - marriage in USA route.
The fiance visa is, generally speaking, faster.
I see no reason to wait. Bear in mind, if consulate receives your I-130 approval, they will not issue a K-3. They will ask you to wait for your green card.
The processing times can be determined from this page http://www.immigration.com/processing-times-and-status-checks. As to expedite, check with USCIS customer service if they have any recommendations.
Wait for instructions from the NVC and then the consulate.
The only way I can think of is driving through from Canada. At the border, his situation should be explained to the CBP officers. It is their discretion to permit or not permit the entry.
I think you will be fine. As I recall, the law only requires you to get married within 90 days. But confirm this information and do check with a lawyer as to whether you may travel or not while AOS is pending. This could be the only issue I can think of.
You could look into a fiancee visa (K-1). It can take a few months.