1) A wireless router is a blinkenlichts box that provides wifi to your house. The box that plugs into your coax is your Cable Modem. Often, your cable company will rent you a combined modem/router, which does both. You can provide your own cable modem, and your own router, and they don't have to be expensive, especially if you buy them used. Spectrum have
a list of authorized routers from which you can choose. Your wireless router is entirely up to you.
2) In a good router, you'll want multiple antennas, good speed, and be from a reputable company (tp-link, netgear, linksys have all been around for a long time). The router model number usually has a number in it (600, or 1200, or 1900, or something like that) which indicates the potential speed that router can support. We used a "600" router in our small business (half a dozen users) for several years, and recently upgraded to a 1200mbps model.
3) Your ISP cares about what cable modem you have, but the rest of your home network doesn't matter to them. You'll need to call them up and say "I've bought my own cable modem and need to activate it." They should walk you through the process. They'll need to know some numbers from the label on the modem, so their system will recognize it as being authorized. You'll need to return the modem/router they've been renting to you as well. Make sure to get a receipt--I've never dealt with Spectrum, but a few years ago Comcast tried to bill me for a modem I had already returned.