Run As system

You enter username and password for the pc your logging in to why does it mention domain?

When it drops open a CMD prompt type
Ipconfig /all
Tracert adobe.com
Tracert 194.119.131.66

Post results

If you can get internet once then create a PIN that doesn't use the net and you still use ma accounts

You say connecting to a modem is it modem or router if it's a modem it very dangerous as your pc can be live on the internet so ant hacker can get to it a router protects with NAT so it's live not...

First move is to run a full diskcheck on the faulty drive second check the connection on it in case they have become loose

from a cmd prompt type

set
is the bin in the path?

If you are a company then a base image is generally the way to go due to time taken to install 10

It should be as simple as going into the bios and settings HD as first boot it's likely it's been set that way to install from a network

Post results from ipconfig /all

Full list how to unhide How To Hide Or Unhide Any Drive In Windows 10

Have you run storage space and removed any install files

The last permission show side with no name. Start with basic
Full disk check with chkdsk
SFC /scannow
Create a new admin account and use that does it work? It could be your profile is corrupted

Corrupt disk can just be file system which check disk can fix the others is the hardware fault with actual bad sector's

Its often down to price 2 gig is best but 300 meg should cover most needs unless you all use at the same time often for gaming ping is important

Follow details here Is it possible to install mint from Windows? - Linux Mint Forums

The WiFi driver is n type very old and slow. Your wrong if you have AC router and get the USB WiFi adapter it's faster than ethernet and quick fix

You can get a ac WiFi USB adapter very cheap with drivers that the quickest solution

One of the photos shows network disabled open a CMD prompt type
Ipconfig /all
Post results from both PC's

Run storage app that can clear things and can be set to run on its own every X days

The adapter can get over 300megs but very small antenna what is the signal strength on the pc if it going through a wall or TV or something between the router that could effect it is it the same on...

If the old drive was legacy and the new bios is set to uefi, secure boot, tpm then it wont boot you have to set the bios the same as the old pc

Is that the only account? Do you know the password? There isn't a simple way to get the pin down you need to get to documents?

To best advise you we need details from post 2 or we are guessing

A usb adapter is £13 or less just plg in and its done...