1. They can, but unless they have a branch/office in that state or unless they partner with a mortgage broker or other mortgage entity in that state it's unlikely. Mortgage law varies enough state by state that few lenders will only do it occasionally - they either support it and have a staff in that state or dedicated to it, or they don't do it at all. See if they have branches or ATMs in the area you want to move to, if they do your chances are better, but ultimately you'll need to ask them.
However, given your circumstances, I doubt you'll have much difficulty getting a pre approval from a variety of lenders, including credit unions, inside the state you move to. Start calling now and get a pre approval from at least one bank/credit union. You don't need to be a current customer to do so, and you need a pre-approval to start the whole process anyway. You don't have to commit to using that provider for the final loan - the pre-approval is enough to make an offer on a house, and you can continue to shop around for the best loan after the offer has been accepted.
6 hours is too long a commute, though, so you'll need to show you have an active job in that area, or a solid job offer.
2. Even though the government FHA program supports a wide variety of loan types, banks prefer loans that are less risky, and as such pre-existing homes are the least risky. Manufactured homes (ie, "mobile homes", not pre-fab) are assets that depreciate, like cars, and thus the value of the property declines, and they don't like that either - they can't easily sell the place if you ditch the loan. Empty land often develops into a home, but a lot less often than you might think, and so that's also more risky. If you want an easier, faster process you need a home on the land, and you don't want to build one or buy a manufactured home.
3. Search for some, and call at least three of them with these and your other questions over the phone, and hand the phone to your SO and have them ask more questions. You'll pretty easily be able to tell which one will mesh well with you two and be easy to work with. They will require a contract pretty early in the process, but you should feel comfortable before they do. Ask if they also represent sellers. We've worked buyer's agents as well as regular and each time we've been happy with the outcome, so don't fret it too much. Buyer's agents don't actually care what home you buy - their income depends on how much you buy, so they're going to push you towards homes at the outside edge of what you can afford just as much as any other agent type will. You will need to use them for what they're good for - lining up inspections, doing the negotiating, understanding the laws, etc.
For our current house neither we nor the seller used agents at all, but we've been through the process enough and are self-motivated enough to do the work. It's not that hard - but they'll protect you from the biggest mistakes.
Just don't make the assumption that they will do what's best for you without also taking into consideration what's best for them, even though they are a "buyer's" agent. That only protects you from one small conflict of interest, leaving you open to all the others.
4, It takes at least one month from the time an offer is placed until you can move in. Sometimes longer, very rarely less. So working backwards, you want one week to move, so you need to get the keys a week before your lease is up, and you need to make an offer 6 weeks before your lease is up (rounding up). You need a a few days to a week for pre-approval. This is still a somewhat quick timeframe given today's market, you might want to plan more than 6 weeks from the offer, and thus start earlier.
You can generally find something "wrong" enough with the house during the inspection that you can get out of a contract after the offer, but then you've lost a month (and thus have to pay another month of inflated rent) so you really should plan on doing the house hunting and research yourself. Even a good realtor will only want to show you a handful of houses before they get tired of spending time with you looking at them. We spent a lot of time on zillow with a wide variety of searches, and we watched houses pop on and off the market. Watch long enough and you'll get an idea of how much a house is really worth based on whether it sold quickly or not. Learn to spend time looking at nearby houses, how recently were they sold and for how much (the biggest benefit of zillow, imo).
You will, however, want to see the house before you put an offer on it. Pictures and text can never do it justice, positive or negative, so you've simply go to go out, which is probably the biggest chore for you, being 6 hours away.
Yes, you can sell the house and buy another one, but if you live in a house less than two years in a good market you're almost guaranteed to lose money due solely to closing costs, and in many cases that may be 5-7 years.
So whatever you choose, make sure you choose carefully enough that you can live with your decision for several years.
Thus you should plan on spending a month just looking online and familiarizing yourself with what's available, what the area is like, the schools and businesses nearby, etc. Spend at least a few days actually viewing houses.
Houses will be snatched up when you're wavering, and you're going to feel a lot of little pieces of regret - ignore them. New houses enter the market all the time, you can't let the feeling of "the one that got away" force you into putting an early offer on something that has significant problems just because it might be the best you can get.
So as a rough guess, I'd plan 10 weeks minimum. Get the pre-approval now, interview some buyer's agents, and start stalking the market on Zillow or a similar service now, none of these lock you into anything, and this process will quickly bring you up to speed regarding a lot of your existing and upcoming questions.
Also, be aware that the FHA loan isn't the only one you should look into. There are a variety of federal home lending programs. We used a USDA rural development loan that happened to apply to the house we wanted and it provided significant benefits and advantages for our situation.
