As someone who is currently studying to be a pilot (22 hours of flight time so far) and who has been around all sorts of airplanes all my life, including ultralights, LSAs (as long as they have been around), classics, homebuilts, kitplanes, original designs, warbirds, etc. I'm as close as you can get to being an expert in this field without being part of the FAA.
First point, homebuilt, kitplanes, and original designs are all under the "experimental" class of aircraft, this means it can't be used for commercial flight (paying passengers or cargo for hire) but this does not restrict recreational flying with passengers or cargo. Once your plane is completed and has gone through the inspections and tests it's signed-off by the FAA and it's free to fly. Flying into controlled airspace ( B, C, D, etc) simply requires the proper equipment on board (transponders, etc), same with flying in IFR conditions (though the pilot also needs IFR rating on their license). If you think that this is a bottleneck that is used to keep the majority out, just go to the annual EAA fly-in in Oshkosh WI, every year last week of July. You'll find at a minimum 1/4 of the airplanes there are homebuilt airplanes, you will even find original designs parked here and there shined and polished and being shown-off by proud owners/builders. The restriction by the FAA is non-existent, they merely regulate and ensure the safety. Judging by the rest of the nation, the FAA is one of the few government agencies that is actually working fairly well to promote freedom and independence.
Second point, yes you need a pilots license to fly. If you want to fly an LSA you can get a sport-pilot license, which has reduced cost (at the expense of increased restriction). Yes it costs more then a drivers license, but by the same token, you driving a car without a license is no less illegal then flying an airplane (other then an ultralight) without a license. Costs for pilot licenses vary, as the majority of the cost is airplane rental and paying the instructor, right now I'm budgeting about $6k for getting my license, that's airplane rental, instructor time, renters insurance, exam costs, etc. That's a lot of money, and it's taken me a while to get that much cleared for the task, but it's not an impossible sum, nor is it an impossible goal. I could go for a sport pilot license, it requires about half the time in the air (20 hours instead of the 40 hours minimum for a pilots license) but I decided to go for the full deal.
Third point, why would you want to? You call up the FAA and tell them you are building an airplane and they'll say "ok, let us know when it's done, oh, and here are some resources to help you out.". You call up the FBI and they'll say "that's nice" and hang up. You call up Secret Service, CIA, TSA, even the IRS and they won't care.
As for the explosives angle, that's a completely different subject and doesn't have anything to do with aircraft. While the 9/11 incident was spectacular it's a fairly isolated incident, most explosives you hear about are driven there in cars or trucks. Explosives themselves can be made from a host of compounds, many of which can be procured just driving past a farm or two, or visiting a hardware store. Other then the TSA playing patty-cake with your privates there is no reason for flying and explosives to be in the same discussion.
If you have any questions or are interested in learning more, just come to the Oshkosh fly-in, or any local fly-in (call your local small airport to find out when the next one is happening, don't bother the big airports, they are too busy with airliners) walk around, talk to the pilots, talk to the people there, and go for a ride (there's usually at least one giving rides at the local fly-ins). You'll find great people, beautiful airplanes, and frank discussions about the realities of flying in the USA and why these people have a big grin on their faces ever time they leave the ground.
