I hope I'll be able to help you out a bit. I'm also new to Mint and loving it
1) open Synaptic Package Manager and check if you have gnome-terminal installed (I think that's the app that might be simply missing on your system). If it's not installed install it then.
2) What I do with .mint files is to download them to the desktop and double-click them once they are downloaded. Then I click on Install in mintInstall and wait until it's done. It'll download and install the necessary package(s).
3) There have been a few updates and some of them for the mintInstall app so you might want to get them. On my system updates come automaticaly (I thought it's so by default). You can use Update Manager in Administration.
For me mint run perfectly well out of the box so I am not sure if I gave exactly the answers you were after. Nonetheless I hope it's helpful