Falls Church eh??
It's a very diverse area, ranging considerably in price. Also, very densely developed, meaning that there are not a lot of affordable homes that have available lot space for a sizeable garage. Especially if you require a flat lot, driveway access, sight lines, etc.
Close in to Metro - way expensive. Now if you were to move further out the Dulles Toll Road... But you will curse that DC commute every day!
Permitting in Falls Church is not any worse than the surrounding counties, and in fact used to be one of the easier municipalities to deal with. Used to require only two sets, stamped and signed, of prints for permit applications (other than for restaurants). Last phone number I had Falls Church Public Works was 703.241.5080. As with any jurisdiction, if you approach this half- a**ed, you will pay the consequences. Any experienced Contractor should be able to walk through any permit drawings (simple structures). Note that this is not the same as zoning.
The difficulty will be meeting established lot setbacks, as well as building heights. And if there is any suspicion it will have commercial applications you will be denied.
There are several "design-build" firms that advertise regularly in that area. That would be one approach. There is a lot to be said for a turn key project. Another would be to hire an Architect for a set of plans, and then solicit bids from approved Contractors. Architect can furnish names of people he has had experience with.
As to a ball park cost, anywhere from $60-80/sq.ft. for a basic garage assuming that there are no extenuating circumstances in construction. Anything much less than that would have to be prefabricated (might be worth looking into - several firms out past Middleburg do that, or there are national companies you could try), or a really starving Contractor. Another approach might be to find a skilled carpenter that could serve as a working foreman for the job and supervise any sub-contractors. Any price that sounds too good to be true will be exactly that. Read some of the stories about garage construction on this very forum!
Don't automatically assume that you will be able to park your trailer in the driveway. You may have to store it inside or off site.
Any options will also raise the price so consider seriously how much you would require plumbing in the garage, or whether you will really need an epoxy floor.
Good luck with your search.
As always, offered only as opinion