So as the title says, I'm trying to put a Node Network I made already into an existing Subnetwork.

What I did is, I created a procedural floor and used parameters from a subnetwork which was empty at the time, thinking it'd be easy to than transfer the procedural floor into the existing subnetwork, keeping all expressions and constraints of this existing subnet.

Try selecting all the nodes.
Press CTRL-X to remove them from the network and place them on the clipboard.
Then dive into your existing subnet and press CTRL-V to paste.

Another option might be to select them all and press SHIFT-C to place them in a new subnet.
Then select the other subnet and edit the parameter interface. Select all those control parameters and do something similar by copying those parameters to the clipboard and pasting them into the other subnet's parameter interface.

just cut and paste the nodes into the subnet…
you'll have to correct the expressions by hand.

Ah okay, I thought there was some kind of trick! I put everything into a new subnet, including the subnet I already had, and then copied all the parameters!

you may not want to work this way…
I'd recommend:
making a subnet
RMB on it and create a Houdini Digital Asset
go inside and start making you network
whenever you want to make a control - make it on the HDA

in the end you'll have an asset - not something that you'll have to mess around with and fix later.