I'm a licensed ham, but haven't done EME (but have read some about it), so I'll try to limit my answers to those areas where I have some knowledge.
First, getting a license isn't very hard. If you're reading this board, you probably have most of the electronic theory background already, and that's what most candidates find the toughest part. Since you've said you're in the US, I'll go out on a limb here and say that getting a license is the easiest and cheapest part of the journey you're planning (other country's license requirements may be harder). But getting a license does require taking an exam, and taking an exam requires finding a site and time where an exam is offered. After the exam, it will take a few weeks for the paperwork to be processed. So try to get that out of the way ASAP. Go to
http://www.arrl.org/getting-licensed and poke around to find out more info. It would be a shame to get to Phase 1D, then start looking for an exam session, and find out that you were a week late for an exam session, and the next one in your area was several months away.
Practice exams are available on-line from various sites. Take a few until you are solidly passing.
Key Question: What license(s) will be needed for Phase 2?
Most EME work is done in the VHF/UHF bands, where a technician has full privileges. So the most basic license will do the trick. However, once you're in an exam session, you can take as many tests as you want for the one fee you paid, until you fail an exam or pass them all. So you might as well try for the general and extra exams while you're there. These license classes open up lots of opportunities on the HF (shortwave) bands, where you can bounce signals off the ionosphere and talk around the world. It's a different niche of Ham radio, but an interesting and popular one.
is it fair to assume that Phase 1A and 1B could both be done with a Yaesu FT-60R (or lesser radio) and a handheld or small tripod mounted antenna?
Phase 1A/1B, and 2A/2B, yes. But it's probably a dead end as far as EME.
The FT-60R is a nice radio, I've got two myself. But it is a low powered FM-only walkie-talkie. Plenty of people use them for satellite communications, with upgraded antennas. You can even talk to the astronauts aboard the ISS, if your timing is just right and you have a bit of luck. But a problem is that it won't send or receive the modes most useful for weak signal work. My understanding is that most EME is done with SSB, Morse, or various digital modes, and an FM-only radio won't do any of that. Someone has probably done EME using FM, but it would require significantly more antenna and power than the other modes. I believe what you want is an all-mode transceiver that covers the bands you're interested in. There are various possibilities here, but I don't know enough to recommend specific makes/models.
There are hams who thrive on a challenge and like to do something most people would call impossible. You could probably get a nice writeup in QST (the ARRL's monthly magazine) if you managed to make an EME contact using an FT-60R. The antenna and amplifier required to accomplish such a feat would not be trivial or inexpensive.
Key Question: Is it correct that all of Phase 1 can be done (in the U.S.) without a FCC license?
Yes, there is no license required for receiving, period. Furthermore, it is legal to own transmitters without holding a license, but it is not legal to transmit with them. So, at least in theory, you could buy a transceiver that you would eventually use for phase 2, even before you got your license. Some sellers might not be eager to sell to you before you were licensed, however.
In Phase 2A it would seem possible to send and receive my own signal. Can sending and receiving my own signal be done in Phase 2B and 2D, or is the nature of LEO and EME Transmit/Receive such that it is unlikely to receive your own signal?
For EME, you are likely to receive your own signal, like it or not. There's a bit more than a 2 second delay for EME round-trip. That's plenty of time for a transceiver to switch from transmit to receive and to begin receiving.