One thing that people commonly misunderstand is the geographic reach of patents. US patent law is only going to apply to things that you do in the US (or import into the us), and you need a US patent for it to apply at all. If you have a US patent but no UK patent, infringing activities in the UK aren't impacted. If you have a UK patent but no US patent, infringing activities in the US aren't impacted. Because of differences in patent laws, it is not uncommon for the actual enforceable part of the patent (the claims) to be different in different jurisdictions.
So taking your questions in order:
(1) "How would using the equipment of others .. effect apply for a patent"? The simple answer is that it probably wouldn't. For example, if I use a Dell computer to model chemicals and file for a patent on the chemicals, Dell doesn't have any claims. The more complex answer is that you have to pay attention to the contract terms with the equipment supplier. The even more complex answer is that government involvement (at least in the US) can impact the patent process (particularly if US government funds are used).
(2) "If I was to tell someone about my invention and they filed for a patent, what would happen? Would it be my fault, or could I sue?" The US recently switched to the same "first inventor to file gets the patent" system that the rest of the world uses. If you tell somebody else about your invention, even if they file first, they aren't an inventor so they shouldn't get a patent. But what should happen and what does happen aren't always the same. If they claim to have invented it, how is the patent office to know they are lying? The safest thing to do is to file first and talk about it later. If you can't do that, then at least get a good non-disclosure agreement in place. Could you sue? It depends on what you are after. If you just want to invalidate the patent, you could seek a reexamination. If you want damages for theft of your idea, you absolutely need to talk with a lawyer about the specifics of the case and figure out if you can sue.
(3) "If a company made my invention without my permission, what would happen?" The easy answer is "nothing would happen". You could have a patent that exactly covers what the infringer is building and still nothing would happen. A patent doesn't give you the right to build something -- it only gives you the right to stop others from building it. So you would have to sue the infringer and seek a court order enjoining them from continuing to infringe (and/or damages). If your invention isn't the subject of an issued patent, you don't have a lot of recourse (although if you disclosed the invention to them under a non-disclosure agreement, you might have a breach of contract claim; there may be other claims as well depending on the specific facts). This would be a situation where a quick talk with a patent litigator would be helpful.