1. Yes you can.
2. TSA clearance(for single, multi and Instrument) are definitely required and this can take a month at least.
3. No you have to do your recency/checks on the multi you wish to be endorsed. Unless you're willing to do a minimum of 10hrs on the duchess, you're better off on the Seneca.
4. Current waiting times for issue of a license is an average of 3.5 months.
Hope this helps.