Well, my favorite use of a computer is to compile new apps, but I know that's not everybody's idea of fun.
But you can do pretty much anything with it.
Personally, I prefer not to install software unless I really need it, otherwise things get really cluttered. (So I'm usually asking the opposite question, "what utilities can I live without?")
I should probably leave these questions for someone more experienced; but, as a start, you can probably just use the "sharing" support built-in to the OS. If you open up System Preferences and select "Sharing", you'll get a panel with a variety of options on the left-hand side. One of those options is "File Sharing"; if you click on that, you'll get a list of folders you can share over your network. Make sure the box in front of "File Sharing" is clicked; and, if your backup disk is not in the list of "Shared Folders", you can add it by clicking on the "+" sign at the bottom of the list. (Also, you'll need to give read and write access to some user; you can then log in as that user from the MBP.) Finally, take note of the message in fine print above the list of shared folders; it'll say something like "Other users can access shared folders on this computer at "afp://nnn.nnn.nnn.nnn".", where the nnns are numeric values. You may need this info later...
Once this is done, you can go to the Finder on the MBP, and from "Go" in the menu bar, select "Connect to Server...". If it already found your Mini, its name will be listed in the lower box of the dialog that pops up. If not, you can type the "afp" value into the "Server Address" box at the top of the dialog.
After that, you should be able to see the remote drive on your MBP. You can then access files from it or store files to it, or use it as a backup drive with "Time Machine", or whatever...
Yes; this feature can be controlled from the "Energy Saver" panel in the System Preferences.
Yes; this feature can be controlled from the "Users & Groups" panel in the System Preferences (choose "Automatic login" for one user).
