Hi Carlos just to get some clarification in your questions.
The Cypress team plan is related only to the Dashboarding features. This is not a team plan to use Cypress.
And when you talk about installing Cypress on a server what do you mean with that. Normally Cypress is started from a coding project. Here Cypress will download binaries to your machine and will be used to execute your tests.
So to answer it shortly yes you can install Cypress on a server/machine and yes you can direct it in the config to store the binaries on a generic place for all users, but i'm wondering why would you have multiple users on the same server?
And to answer your last question Cypress is a tool to run automated testing. You will not be doing any manual testing with Cypress.
If you are really new with Cypress I would suggest to read the Getting Started: https://docs.cypress.io/guides/getting-started/installing-cypress
Or watch some tutorials on youtube: