It's a good plan begin with as simple a network as possible.
To start, I'd actually recommend connecting both your laptop and your cRIO directly via Ethernet to your school network and confirm you can run your Driver Station successfully.
Then you are starting with something simple that you are sure works and you can add one new wrinkle at a time:
- laptop->school router->cRIO (Ethernet all)
- DAP dropped in via Ethernet (laptop->school router->DAP->cRIO)
- Laptop to wireless (laptop/school AP/router->DAP->cRIO)
- DAP to wireless (laptop/school AP/router/DAP->cRIO)
Make sure the DAP is set to bridge mode, not AP.
The multiple switches won't matter.
I oversimplified my home network, it actually goes:
AP/router -> 16-port switch -> 8-port switch -> cRIO