When you receive an email with an attachment, you can do a Quick View to look at the file. This is an Apple thing in the Mail app, where you can just look at the file. However, you can't edit it. So, Apple invented the "Open In..." feature.
Press and hold the attachment. You should get a pop-up titled "Open In" that will offer you a list of apps you have installed tht have the capability to open and/or edit that file. You select the app you wish to use and it will open with your attachment. Do note that if you don't have an app installed on your iPad that can open an attachment you've received, you will not get that "Open In..." option. You'll have to get something from the App Store to view that file.
When you open a file with a particular app, that app makes a copy of the file. So, you can always open that app and find the file (as long as you haven't deleted it from that app).
As far as music and movies that you downloaded, it works the same way. You have to use an app to do so and that movie or music file will be within that app. However, usually, you are not downloading music and movies from the Internet unless you know it and are in an app specifically designed to do so. You usually are only streaming, which is live action and is not being saved any where.
You do have access to your personal files. However, it's opposite from what you are probably used to from using a personal computer. On a computer, you can find a file and the computer will determine which program to use to open it. On an iPad, you open an app (program) and then determine which files it has you can view.
Hope some of this helps or clarifies. Let us know if you need more.
Just thought I would mention Goodreader app since it seems to satisfy my Windows friends (I'm Windows too BTW) who want folders and being able to move files, etc. You can do all of this, open most files, backup in Dropbox--lots more. You might feel comfortable with it, being able to see all your files in one place, being able to organize them, back up in Dropbox AND Icloud.
Yes, it's very feasible. Two apps that I've used are OPlayer HD ($4.99, USD, but you can get a free Lite version to try it out) and AcePlayer ($2.99, USD). Between the two, I've not had any issue opening an .avi file (or other video files).
Of course, these are two I'm using - no endorsement intended and YMMV. There are plenty of video player apps in the App Store. Have a browse in there...
I vote for OPlayer Lite since I haven't had any issues with it yet. Worth trying "free" first LOL