Update: I found the solution to #1. Even though date types are returned from the REST API in Unix "seconds-since-the-Epoch" format, they must be supplied in search queries in 'yyyy-MM-dd HH:mm:ss' format. E.g.:
GET /Api/rest/1.0/data/contacts?search=C_DateModified%3E'2014-02-05%2015%3A36%3A58'
I'm still looking for help with #2 and #3 though. Thanks!