I'll try and answer these as best as I can.
Just wondering, are those things not covered by law in the US (or does it vary from state to state)?
It varies from state to state. There are very few federal regulations regulating the storage of firearms. The only one I am aware of, and it's been so long I can't even be totally sure if its a federal or state law, is that if you are living with a person who is prohibited from possessing firearms or ammunition, then it must be kept inaccessible to the prohibited party.
There have been movements pushing for safe storage laws, but I am, frankly, skeptical about them. If you're curious as to why, I will be happy to explain later but for now I want to try and answer some of the differences between the US and the UK, at least from a Yank perspective, as objectively as I can.
I think I'm right in saying that in the UK it's illegal to let someone borrow your gun (although I'm sure it happens) but I could be wrong about that
Likely, but I can't say with any degree of certainty as to United Kingdom regulations regarding the loaning of firearms.
but I am fairly certain that you MUST keep you gun unloaded and dissembled in a gun safe, which must comply with certain specifications, when it's not in use.
Some states have that regulation, the above mentioned "Safe Storage Laws." California, Connecticut, Massachusetts and New York, I believe, are the only states that have them. All new firearms sold are required to come with a locking device and these locking devices can also be obtained at local police departments and through the National Shooting Sports Foundation cheap or, in some cases, free.
Also, if you are going to be away from home for longer than a certain amount of time (like say if you're going on holiday) the gun must be handed in to the police for the duration of that period.
We don't have have those requirements here, but I'd love it if that service was available on a voluntary bases.
Plus if you apply for a gun license the police will send someone round to your house to make sure your gun safe complies with the law
We don't have that here, no, and that would also be a very sticky legal situation if they tried to institute it.
and also to check to see if you are an obvious nutter who shouldn't be let near a soft cushion never mind a firearm (obviously nutters do still get licenses, they aren't sending psychiatrists to your house, but maybe they should).
When purchasing a firearm from a dealer anywhere in the United States, you are required to fill out paperwork to run through the National Instant Criminal Background Check System (NICS). This is known as an ATF Form 4473. After you fill this out, the dealer calls the FBI and they run a check on your background for any disqualifying factors, such as felonies, disbarring misdemeanors, and whether or not you've been adjudicated mentally incompetent or defective. It usually takes approximately five minutes for them to find out whether or not you're a prohibited possessor. The feds will then either send back to the dealer a Go, No Go, or a Hold. Go and No Go are pretty well self-explanatory. A Hold means that something pinged which, while it wouldn't outright disqualify you, it does warrant further investigation. A Hold can only be effective for between four and five days while the FBI investigates further. If the investigation isn't complete within that time, then it falls to the dealer to decide whether or not to go ahead with the transfer. If the transfer goes through, but the investigation later turns up something that would disqualify you, then they will send law enforcement around to your home to take possession of the firearm.
Are none of those things also law and part of the application process in the US?
Some are, some aren't.
If so that seems like MASSIVE oversight that should be addressed...none of those things imo would go against the right to bear arms, at least not in any unreasonable way.
Mmm...yes and no. Because firearm ownership is a protected constitutional right, you have to be able to have laws that would not violate that or your other rights. For example, earlier I said police inspections of your home would be a sticky thing legally. That's because choosing to exercise one right, your Second Amendment right, cannot exclude you from protections provided by another right, your Fourth Amendment right. The Fourth Amendment protects you from searches without a warrant for probable cause that a crime has been, or is about to be, committed. Treating the desire to own a firearm as grounds for a search constitutes a presumption of guilt and that is something that doesn't (often) fly in our criminal justice system. Some might try to argue that an inspection isn't a search, but it would be a requirement that we don't have when exercising any of our other rights and that, in itself, would render the law illegal.
I mean there's already some restrictions on that right in the US - for example it's pretty damn difficult to get hold of stuff like miniguns legally,
This is actually a fairly common misconception; it is actually fairly easy to get a hold of a minigun, it's just fuck all expensive. For this, I'll use your example of a minigun, which, I presume, you're talking about a GE M134 Minigun.
The first thing you'd need to do is actually locate one that was made prior to 1986. After that, you have to take it, fingerprint cards and two passport photos to a Class 3 dealer and fill out an ATF Form 4 and, if you're registering the firearm as an individual, a Form 5330.20 to prove you are a United States citizen. The dealer will then hold the weapon for you while the paperwork is processed.
The dealer will submit the paperwork and if all is clear, you'll then have to go get your Chief Law Enforcement Officer's signature stating that having the firearm won't violate any laws. This is largely a hold over from when the NFA was passed in 1934 and hadn't been updated after the introduction of the NICS system; and some CLEOs will refuse to sign off on the paperwork, even if having the firearm is legal just because they, personally, don't like them.
Remember when I said it was fuck you expensive? For every Class 3 item you purchase, it requires a $200 dollar tax stamp on top of the cost of the firearm. The last transferable GE 134 Minigun I saw would set you back $215,000 dollars, so if you can swing that, I think you can find the $200 dollars somewhere in the couch.
(NFA process: http://blog.adamsarms.net/blog/how-do-i-get-an-nfa-tax-stamp
and civilians aren't allowed to own nuclear bombs.
This is hyperbole and makes baby Jesus weep tears of semen.
So clearly people are ok with some level of control.
Some people yes, some people no. And the level of control varies person to person. For example, on this board regarding this topic, I am probably one of those that would land somewhere on the right (conservative) side of the bell curve while, I presume, most of the rest of the board would land somewhere on the left (liberal) side, and yet I am probably one of the most overall left-leaning people here, at least I was the last time I saw everyone posting their political compasses, so I try not to paint with too broad a brush when regarding firearm ownership.
I hope I've helped answer some of your questions.