amd21

You cannot see all of the applications listed under AdminCP -> Applications (under the heading Site Features)? If not, you have not uploaded all of the files and folders as Marc has said.
You need to upload the entire contents of the `ips_xxxxx` folder (not the folder itself) to your forum main directory, overwriting files and folders when prompted and making sure all files transfer without error. You will then go to your site adding the following to your URL `/admin/upgrade` and login with your AdminCP credentials. The upgrade system will scan your install to confirm that all files are up to date before proceeding.
Let us know how you go. If you still encounter issues, it may be best to consult someone from the Providers Directly to help you out, or you can get in touch with your hosting service.

As Marc said, you have to upload those files via FTP to your server.
Ignore that "Install Application" popup. It's not what you need right now. You use that only when you have a TAR file to install/update single applications. IPS doesn't provide individual TAR files for their applications.

No, you would be uploading files via FTP. Not via the admin CP. I have to be honest here and say I really really would advise getting someone to assist you with actually doing this if you are unsure, as you can break your site. It also may be worth at taking a look at our cloud solution rather than self hosted, if you are not sure on hosting of your site

You don’t download single apps. You download the full software package. When you download it, the system knows based on your license what files to include. Upload everything to your site. 

You would upload the full file set to the location of your community on your server, overwriting what is there. Then go to ACP -> System -> Applications to install your missing applications.

Creating a test site is the same as making any other site. The only difference is the key you use, as mentioned above

Yes, you want to make a completely separate installation. This means uploading a copy of the software files again and also using a different MySQL database so that you don’t risk impacting your live installation. 

I personally would suggest using a separate hostname like test.mydomain.com as well.  That way if you need to delete/reinstall your test instance you won’t have issues with the license key saying it is already in use.