Following this guide (linuxtechi) and got Mint Cinnamon 22.2 (Zara) - yes the iso is verified, created bootable USB with Etcher, and after the screen where I input user details and password, well along the install process, got a fatal error (screenshot) :
Unable to install GRUB in /dev/sda
Executing 'grub-install/dev/sda' failed
This is a fatal error
Here is a screenshot of the GParted, fdisk, df, lsblk and what re-running the install now looks like.

Restarting (after pulling out USB) , instead of going to Windows 10, goes to a black screen that says

This is not a bootable disk. Please insert a bootable floppy and press any key to try again ...
Turn it off, back on with USB, at least I can still boot from there into USB-space mint for now.

I'm guessing this has something to do with some peculiar hardware/BIOS settings than the fault of Mint. Worth mentioning that this is a refurbished 2013 HP box (order excerpt) , with a windows 10 digital key license, and that upon every fresh boot up (first time ever to now) it shows a black-background screen that says

The preboot authentication application cannot be found.
Press any key to attempt boot without authenticating.
so it's very possible something is bork'd from the get go or else it has some very unorthodox settings to begin with.

Can confirm running Mint off the USB is fine. The screencaps are sent from Firefox within, and I'm super happy with how clean everything is. Just not sure what to do now. If it's complicated, I can try sending the box back for refund and try again with another brand new box. But now I'm shy about trying dual boot on my regular desktop.

Questions after looking around for help:

1. am I booting from UEFI or Legacy? I don't know -- how do I check ?
2. Fatal error installation hard stop: is there a way to access a log of what happened ?
3. How do I boot back into Windows 10 for now ? use GParted to delete all the "new" partitions and try booting again?

Edit: gave up . It's now a Mint box. Goodbye Windows you can kiss my dust

Background: I started daily driving Linux (specifically Mint) several months ago and for the most part it's been great. Some weird hiccups occasionally but nothing I can't handle/deal with. When doing my research to set the system up for the first time, I decided to go with 30GB for the / partition and ~220GB for the /home partition (the other half of the drive is for Windows 11 and the various essential tiny partitions). For a while this seemed to be fine, but lately I've been starting to get warnings when performing software updates via the Update Manager that the / partition is running out of space. I think it peaked at maybe 90-95% full a few weeks ago, at which point I started doing some research and cleaning up a bit (apt autoremove, deleting old logs and kernels, etc). I was able to claw back ~4GB and kick the can down the road, but now the warnings have returned and I'd like to handle this properly. I'm working from the assumption that I simply made the / partition too small and intend to double it by giving it some of the Windows space.

My question: How do I know if this is expected or if I've been doing something wrong? Is 30GB indeed too small on modern Mint, or should that have been enough? I know it partially depends on what all I actually do with the computer, but I really don't have very many applications installed (aside from defaults of course) and only four of them are Flatpaks, which I read tend to be larger. If it helps, the Disk Usage Analyzer reports that within /, /usr is taking up 13.2GB (55%), /var accounts for 9.1GB (38%), /opt is 1GB (4%), and everything else is <1% each.

Thanks in advance!