That can be a good solution at least if electricity costs are not a big deal. If power is expensive in your area, it might be worth to buy something more power-efficient, like a raspberry pi (assuming they’re not completely sold out right now).
No company is just going to host a server for you for free, a (virtual) server for running nextcloud will cost you at least a few bucks a month. As others have already said, you can run a server at home on your own hardware, but this is also not free (hardware cost, electricity, etc.) and you will additionally have to deal with any hardware issues & replacements yourself.
I think it’s a fairly standard feature. At least Protonmail also supports this kind of “alias”.
Are you using the Jerboa client? I think they recently introduced an option to open links in a private tab which is on by default for some reason. It confused me too until I found the setting.
I’m using nebula to remotely access the raspberry pi in my home network and it mostly just works. The dual setup for nextcloud might be a bit more tricky, at least if you want to use HTTPS. You’ll probably have to set up a reverse proxy in Nginx for at least one of the routes, since they need different certificates (although since Nebula already authenticates and encrypts your traffic, HTTPS is probably not necessary there).