Hi htc_one, and welcome to AF
Root is the linux administrator account, so rooting is the process of gaining administrator access. In itself it changes nothing about the phone's operation, it just gives you the option of changing things you couldn't change before, if you choose to do so. The analogy is between using an administrator account vs an ordinary user account on a PC.
Flashing is a generic term for the way you install system software - the name comes from the fact that this is kept in "flash storage". The same name is used whether you are installing via a custom recovery (as we would for custom ROMs), or using an official HTC update package.
Installing a new launcher won't cause any problems - it's just an ordinary app. A launcher is an app that provides desktops, docks and an app drawer, and the HTC Sense launcher is no different. You can install multiple launchers on your phone and change between them as you feel. I'm currently running Apex on my phone just because it's more flexible than the Sense launcher. The only thing you loose is that HTC's own widgets only work with the HTC launcher. You seriously do not have to worry about damaging the phone by installing apps like this.
Root Explorer is just a file explorer app, except that it's one that's capable of exploring the whole filesystem, including system files. Without root you won't be able to do any damage with it. With root and S-Off you can, if you choose, use it to add or remove system files, which of course could potentially stop your ROM working. Of course the same is true if you use the administrator account on your PC - if you decide to delete things without knowing what they are needed for you could make your PC unbootable. As long as you don't use it to alter things that you don't know can be safely altered it's perfectly safe. The same is true for any app with the capability of altering system software - just don't fiddle with things unless you understand what you are doing and you'll be fine.
If you choose to root, there's one fundamental rule to follow. Your main tool is a custom recovery - a replacement for the HTC recovery program which allows you to do a lot more, including clear parts of the phone's data, install software (eg flash custom ROMs), and crucially to make complete system backups. So the rule is that before you make any
change to system software, even as simple as rooting in the first place, you take a backup from the custom recovery first (often called a nandroid backup). That way if anything goes wrong, you can just reload the recovery. And I would apply this rule to using Root Explorer or Titanium Backup to remove bloatware as well as to flashing new ROMs - if you remove something that was important, as long as you have a nandroid it's very easy to fix.
Ah, I see Frisco got in while I was typing all of that