Can I Trash These Files?

Most things in the Downloads folder can be deleted.

 

Yep. After you've mounted the .dmg or .zip files and copied whatever was in them, you can freely delete the .dmg/.zip files and eject the mounted images.

 

Pictures are usually saved in the Pciture folder in your User folder.
There is a special folder that si calles something library or iphoto library.
It is just a folder with your pictures and some thumbnails in it but finder treats it like a package taht you cannot open.
That is Apples way of keeping unknowing users from messing up those libs that programs like iphoto rely on and iphoto probaly starts crashing or cannot find stuff if you move it.
I cannot stand finder for that and other behavior. I use Pathfinder which can open anything and gives many more options in sorting and stuff.
Aperture I don't know I don't use it but I think it has its seperate library but access to the iphoto one. I am guessing it only copies stuff that you edit with aperture and keeps a seperate copy.

I doubt your pictures are saved more than twice.
dmg more or less are the similar to windows installer packages. Once installed you can delete it. Technically you can launch an app directly from the dmg but if you did that the app is not in your Applications list.
zip,rar, 7z, tarballs it is all the same in Windows, Linux, OSX delete them when you got the stuff in them out.

 

Both iPhoto and/or Aperture libraries may contain your photos, or only pointers to them. Check the prefs in each to see how you're handling import. I find it preferable to leave them in folders I designate rather than importing them into the library. Note that if you had some in downloads, and subsequently move them, Aperture may "lose" them. In that case reimport.

Aperture has lots of options for storage; find a good tutorial and then get a system going and it will be much easier.

 

Learn the unix command line. OSX has a terminal application.
change directory: cd
list contents: ls (the unix equivalent of dir)
cp = copy, mv = move, and so on

Whether or not you want to keep the dmg files depends on you. They are not being used, i.e. they are not needed by any installed programs. However if you want to reinstall a program for which you downloaded a .dmg, you will have to re-download it.