https://www.tosecdev.org/forum/I think this is maybe a troll post, but anyway.
EDIT: Saw tomse's reply when I clicked to post this, but I spent a few minutes on this so I am not killing it.
Don't be offended, but I think you are in the wrong place for an "elementary school emulation training".
You need to find a more generic emulation oriented forum and ask this kind of questions, then spend some time (days, weeks, months) to get "used" to how things work in emulation.
That said, I will try to give you a very brief reply to your specific questions BUT each of these replies "hides" a whole... book behind it. So you need to dig deeper to actually grasp what is going on.
1) .dat files (in our context). The files that store "rom" information, including hash numbers (to verify validity of a file), proper naming of the filename etc.
2) .cue files. This is not related to emulation. One of the methods of ripping CD/DVD, is a multi file format (starting from just two, .cue being one of the two). Within the .cue file is described what files to expect to have the full "rip" of the disc and some basic details for them.
TOSEC collection includes .cue files because (a) they are not copyrighted in any way and (b) cannot easily be created from scratch if someone has just the disc image.
3) rom files. ROM files in general are called the dumps of various physical disks/discs or even actual ROM chips, needed to actually store software in a format understood by emulators. It is a very generic term and is actually a bogus name for many cases. But people are used to use it.
TOSEC does NOT include actual ROMs. You have to find them yourself in specialized fora/tracker sites etc. Some are more underground than others, depending on the possible copyright status of material stored.
TOSEC just catalogs rips even bad ones.
4) What emulators to use. This question cannot be answered. There are books written on the subject. Emulate what? ANYTHING? Well there is MAME that is the most massive emulation project out there. There are many others (and by many I mean MANY). Many emulators specific to systems (what system are you interested) etc. You need to do your own research on this... badly.
5) You miss all these roms because TOSEC doesn't include roms. You need to find them yourself. You actually don't NEED to. You probably are better off to find few specific roms of your favorite system, find a proper emulator (IF it exists) and try to run those (which is not always a matter of just pointing to the rom and running it - many systems have PLENTY of compatibility settings). Again read #4.
6) RomVault is a tool to verify/rebuild rom sets from dats. You need first to grasp what those two things are (not just by my description above).
7) There are guides for these things but most are not up to date. You really cannot just shortcut to something like that. You need to spend time and see how things work starting from way more basic things. Again, think of your favorite system, google for an emulator for it. Run it, "feel it", see how it works. Remember 2-3-5 (not more) of your favorite games. Find them (cannot give pointers to that, start simply by googling). Try to run them. Find a more generic emulation forum to ask your specific questions (and not questions like the above that whole books are written about - fact not exaggeration, I am actually writing my own too).
8 ) You need emulators or sometimes physical systems to run those "games". TOSEC doesn't catalog only games.
Also youtube may help you.
If you think you just discovered a wealth of fun, just lying there. You are right and wrong.