Can I use the CAT3 cable in my home for internet?
A Definite Maybe...
Harper's answer gives a good overview of the general setup of a wired home network. There are two separate issues in your particular case that complicate things:
CAT 3
CAT 3 is nominally rated for 10 Mbps. It may be able to handle 100 Mbps. But it is not officially designed for that, and there are actual differences between CAT 3, CAT 5, CAT 5e, CAT 6, etc. that make the newer cable types able to handle higher speeds. Forget Gigabit networking on CAT 3, but the vast majority of home users simply don't need that unless they are doing true networking between their computers - for myself the only time I truly need it is when I am replacing a computer and need to transfer everything over the wire (and there are ways around that too)).
The big catch is that if you hook things up with CAT 3 and it can't handle the desired 100 Mbps then there are two possible things that could happen:

Downgrade to 10 Mbps - This is actually fine for a lot of uses. May not work well for 4K video, but probably work fine for most other things. That's still 6x the speed of standard DSL of just a few years ago, and is for each device in the network, not shared like a DSL connection.
Errors. Lots and lots of errors. Get an error once every minute and it is no big deal. Get errors every other packet and it is a different story. A lot of things will go into this calculation, but mostly dependent on the quality of the cable (CAT 3 is all the same, but how good are the connections, the network cards, switches, etc.)

A switch that lets you force 10 Mbps. can solve the problem, but your typical cheap unmanaged switches these days don't have that capability. Many network cards have that capability, if you know where to find it.
Wiring Topology
Twisted-pair Ethernet is based on a hub/spoke topology. A bunch of devices connect to a central location - hub or switch (these days, almost always a switch). If your house is set up for that then each of those CAT 3 cables has the other end in one location - typically in the basement, but could be anywhere.
However, plain old telephone service (POTS) is often installed daisy-chained to save on cabling costs. You might have one cable going to bedroom 1 with another cable connected to that one (call it a splitter or whatever you want) and from there to bedroom 2. While POTS doesn't need even CAT 3, it was quite common 25 - 30 years ago to simply use CAT 3 for everything (why stock two cable types) and a little later to use CAT 3 (especially if you had a few boxes lying around) for POTS and CAT 5 for networking.
Since you have some rooms with 2 cables in each location and some with 1 cable in each location, that indicates a real possibility of daisy-chained cables.