A few questions about ME3explorer. I am tech illiterate so I am worried I messed it up. please help

The site www.nexusmods.com/ is trustworthy. And everything hosted there (up to a maximum size of two or three GB) is scanned for viruses and also trustworthy.
Stuff that exceeded that limit and was not scanned is usually marked with a flag, though I think that is only visible in the download section.

As for ME3Explorer, which one exactly did you download? There are several branches hosted on Nexus, e.g., the one currently developed by Mgamerz.
I only have a rather old version on my pc, and am unfamiliar with that 'manual library' term. Where did you see that?

Also: Most content mods for ME2 and 3 are now in DLC form, and do not need ME3Explorer (it is primarily a mod creation tool), while Texture mods for the entire Trilogy now mostly come as MEM mods - using a different modding and installing tool.
So maybe you could state what you want to in the first place?

Both John's alternative MEHEM and EGM come with their own installers, and install as DLC mod.
Installing those is as easy as starting the installer.

As for the Traynor bathing mod. Do you have a specific one in mind? I've seen one a looong time ago, but didn't bother to check what type of mod it is now.
The page should contain a pointer which type of mod (Texture only, or model changes too) it is and which tool to use to install it.

I have never really bothered to use the MEM tool (apart from what comes with MEUITM), but for ME3Explorer the exists a wiki:
me3explorer.fandom.com/wiki/ME3Explorer_Wiki

Yes, dowload the mods, start the installer, and you're done.
Well, mostly. The new EGM version also has an addition tool for all the mods settings. IIRC that is automatically started after the installer.

You don't need any special membership for nexus. Downloads are enabled, as long as you are logged in with your account. Paid membership just means a potentially higher download speed, and no adds on the site.

Some texture mods, like ALOT and MEUITM come with their own version of MEM (at least they did the last time i tried them). For those you also just have to start the provided installer.

For others you would need to look at the stand alone MEM program (I haven't tried that, so can't comment on it), or use ME3 explorer.
If you use ME3Exploerer, you'd need to scan your install first using the Texplorer. That creates a list of textures it can find.
To install the Textures, you would then start the TPF tools, which uses the list created by Texplorer to find and overwrite textures with whatever you let it import.

Though if you use ME3Explorer you really should take a look at the wiki page i linked. The descriptions there are way better than whatever I can say