^^^1 & 2. Initial (new hire) ASI requires a medical. However Many positions in Staff HQ, Regional Offices and Air Carrier Field Offices (CMO) do not require a medical. ASAP Program Mgr, APM, AAPM positions do not require a medical.
3 Unknown, call the offices and ask about staffing.
4. Your entire career, you will visit OKC at least 1-3 times a year, for 1-3 weeks. First year is typically a few months in office (OJT) then 1 month OKC for initial string #1 then a break (back in office) and finally string #2 which is about 6 weeks long.