1. No and yes.
Fares on Metrorail vary by distance and time; they range from $1.70 to $6.75 for a one-way trip. Fares on local buses are flat rate, but each system charges a different flat amount (e.g., Metrobus $1.60, DC Circulator $1). If you're traveling just within the central area, Metrorail fares will generally be in the $2-3 range even at rush hour.
DC Circulator is free with a transfer from another service.
2. Transfers here are better planned than in Chicago (but not as well as in, say, Stockholm or Hong Kong). There are several stations where lines intersect, and you can change for free between platforms within the station; these are shown as double circles on the map. There's also one between-stations free transfer (similar to State/Lake or Library/Jackson), between Farragut North/West. You can switch on the same platform for free at any station served by multiple lines.
3. You can pay cash for a paper rail ticket or for bus rides, but if you ride Metrorail more than a few times, you should purchase a SmarTrip card -- a smart card similar to the Chicago Card. You can purchase SmarTrip for $5 at any Metro station or CVS pharmacy, and in addition to significant discounts on fares (20¢ on each Metrobus ride, $1 on each Metrorail ride) it's faster and you don't really need to worry about the differing fare amounts. If you register your SmarTrip card, $3 is rebated back, so it really only costs $2. Plus, you can even use SmarTrip on buses and trains in Baltimore and DC's suburbs. It's a good deal.
4. Not all that well. Two frequent bus services ply the route: the Georgetown-K Street Circulator bus route, and the Metrobus "30s" service (routes 32 & 36) run down Wisconsin and M streets in Georgetown and into downtown DC (e.g., White House). The Circulator runs to Union Station, while the 30s run along Pennsylvania to the Capitol.
5. Whatever transit services show on Google Maps on your computer will also show on Google Maps for iPhone. DC Circulator shows up as "DCWE" on my computer, but do keep in mind that the schedules shown are approximate.