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If you have the default version of Mint installed then your desktop environment is Cinnamon. There are also XFCE and MATE versions, but you have to go out of your way to get those. The default file explorer for Cinnamon is called Nemo, so if you haven’t changed it that would be what you are using.
Honestly, I think your best bet is trying Disks or maybe gparted if you like cli apps, and setting a mount point for the device from one of those. Linux doesn’t always like NTFS, but you should at least be able to mount and read the drive consistently, although I have to admit I’ve never used an NTFS formatted external drive, so maybe something weird is going on with that.
The whole apt ecosystem is kind of a mess, if you ask me. Debian stable updates on archeological timescales, Debian testing just isn’t a very good rolling release disto, you’re better off with Arch or OpenSuse Tumbleweed if you want to actually use a rolling release as a daily driver, Ubuntu is a mess of annoying corporate decisions I hate from Canonical, and all the others are all just kind of disjointed in how they try to fix those issues.
My personal favorite is Mint. They just try to make Ubuntu with some classic, boring desktop design and minus the more controversial Canonical decisions, but obviously that’s not everyone’s cup of tea. I dunno, there is no perfect distro, you just have to find the one that for you it takes the least amount of effort to fix. Ubuntu really just kind of makes it a pain in the butt to fix all their weirdness though.