This might be a stupid question... Connecting two terminals by a LAN cable

Ok Im really starting to get confused here, here is a quick diagram I made to demonstrate my setip

I’m saying get your own router that supports DDWRT open source firmware. You can set it up to connect to wifi like an access point and then plug your two computers into that routers ports.

Its the same thing that Johns recommended but instead of using your computer as an access point it would use your router.

Just recommending another option the one @johns recommended certainly might work too and no need to buy router in that case.

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One thing I did for years when first moved in to last flat was just a basic wireless access point which had a client option on wifi settings and this connected to the wifi with just a short cat5 into my switch.
Depends on how advanced you need to get.
That would be simple option or you can do more advanced setup with a DDWRT box where the wireless conection becomes a WAN connection as if you had own broadband and the lan side would be isolated within you area meaning better security.
Either way if they are blocking FTP and remote desktop at the main gateway there will not be much you can do with this type of setup as it will go through there router which is out of your control.
I would say VPN might be an option as traffic within VPN would not be seen by their filters however you would usually still need to pass/port forward VPN traffic at the main gateway/router which is again out of your control and if they block remote desktop I would hazzard a prediction they also disallow vpn.

Teamviewer might be an easy option for remote control as runs on different protocols which I expect would get through and also removes the need for port forwarding which would be required for remote desktop/vnc control. Teamviewer you dont even need to know the public IP as the client connected to their server removing the need for portforwarding erc.

Theve blocked Teamviewer as well… Pretty much any remote desktop conection theyeved blocked.

TBH I think your S**t out of luck LOL
VPN in theory could get you past a firewall so long as you can get a VPN connection through itself as traffic within a VPN is encrypted but since this usually requires port forwarding at the main gateway your probably going to struggle on that one as well…

What kind of premises is it?
Remote Desktop, Teamviewer and VPN are all fairly standard in the world of business and surprised they completely block them full stop.
Have you asked if you can have a port forwarded to your unit?

You don’t need access to the Router in the Red box. You do not need access to the Red Router unless you want to remote into your computers from home or elsewhere via the Internet. The Red Router blocking has no bearing whatsoever on making your setup work between Computer 1 and 2.

Option 1:
Connect the 2 computers using Ethernet cable, Connect computers to Red Router using WiFi, bridge the networks on each computer.

Option 2:
Connect the 2 computers using Ethernet cable, Connect computer 1 to Red Router using WiFi, bridge the network on computer 1.

Option 3:
Buy a Router and connect both computers to your Router using cable. Set up your Router to act as a Client so that it can connect via WiFi to the Red Router.

Option 4:
Buy a Router and connect both computers to your Router using WiFi. Set up your Router to act as a host (for your computers WiFi) and as a Client so that it can connect via WiFi to the Red Router.

Any of these options will allow for sharing the DB or Remote Desktop software of any kind, regardless of the blocking going on at the Red Router.

Its a school, but running a coffee shop for them.