Can I have 2 modems?

You cannot connect two modems (or modem/routers) to a single phone or cable line.

Nearly all computers (especially laptops) have wifi capability these days, so you should be able to connect your work machine via the wifi.

However, if it must have an ethernet connection then you can use Powerline adaptors to creata a virtual ethernet cable between the modem/router and the computer location.

In 45 years in telecommunications I have never been stopped.

There are many ways to run a connection.  HOW depends on  your home, your wiring, etc.. Starting with "is it a cable modem, or a DSL modem?" "owned or rented?"  "Age of telephone wiring?"

If the telephone wiring is new enough, it is trivial to use the telephone cable for an Ethernet connection.

If you want my help, I need pictures.  E-mail me.

how the heck are they going to know if you're using wifi or not?

you can always use a wireless access point.

I think you may be confusing modems with routers. Modems are the interface between your home network and the internet. Routers create a network between all devices on the network and OTHER ROUTERS, if any.

So what you can do is set up a router next to your main modem, and set up a second router next to your computer that connects to the original router, which then is connected to the modem. THis will allow you to run ethernet from router 2 to your computer, while it talks to router 1 wirelessly.