A splitter will attenuated the signal. If it doesn't cause any issues, I would terminate the unused port with a 75 ohm terminator to limit the amount of reflected signal.

The bottom line, it will probably be OK.

I only meant you don't need to match vendors.

Matching vendors is not necessary. I have Ruckus APs and a Cisco 3750 switch playing nicely with pfSense and multiple VLANs, both wired and wireless.

Before going down either road, check licensing for the equipment you are considering.

The long white 'cable' is probably a tube with either a pull string for future fiber or already has the fiber in it. What does the writing say?

A splitter will attenuate signal level to/from your modem. Typically 3-7 dB, depending on the splitter. If you do not have a need to split, the coax, use a female-to-female feed-thru connector. If the signals are already marginal, don't use a splitter. Check the upstream signal level, before moving the modem. If it's getting above 42-ish dB, then splitter is a bad idea. The downstream signal level should be above -5-ish dB if you will add a splitter.

I haven't noticed the S33 getting as hot as some older modems, but I haven't paid that much attention. If it gets hot, then keep it in the open. Personally, I like to see all of the lights/LEDs.

What is the brand/model of the gateway? Is the gateway a router, modem or both?

As u/michrech said, lag-free gaming is not possible. There will always be lag at some point. There are hundreds of posts in this subreddit about lag.

Your friend has the right idea, a dedicated WiFi6 AP wired to your router via a PoE switch. Place the AP in the same room that you will likely game wirelessly.

Do you already own the ASUS RT-AX86U Pro? If so, try it before you buy something else. 'Gaming' is mostly just marketing BS. In the consumer world 'Pro' doesn't mean much, either. The ASUS RT-AX86U looks like it will route 1 Gbps.

Whichever router you pick, disable its WiFi in favor of the 'real' AP.

Try setting your ethernet interface to get its IP address via DHCP. Let it get the default route from the modem, then see if you can access 192.168.40.1.

I took a quick look at what happens in my router when accessing my modem. It looks like the modem intercepts its own traffic. It should have been forwarded to my default router, but the default router shouldn't know anything about the modem. I'll get a packet capture when I get a chance.

Since you have an antenna, I'm assuming that you are getting internet from some form of radio service (WISP or cellular, maybe). What if your family parks somewhere else? Could the car be blocking the signal to the antenna? Does the bandwidth return to normal, if your family turns off their phones?

If you have coax in the two areas, MoCA can provide a wired connection between the router and switch.

If you are strictly using IP addresses in the same subnet, there should not be any name resolution involved. The initiating host will ARP for the MAC address of the host with the IP address that you specified.

If you use ping without the -n option, it will try to resolve the host name associated with the entered IP address. If the DNS query times out or gets an error, ping silently displays the IP address in the host name field instead of the host name. I believe that this is a one and done and should not effect the actual pinging.

You must register the MAC address of the modem with your cable internet provider. If they only allow their modems, you will not be able to use your own modem.

Nothing is wrong with your traceroute to www.google.com.

You have 0% packet loss to the final destination. The mid-path packet loss is due to those routers not responding to ping. Ping is not always a priority.

It would be a better test to traceroute to the actual servers that are problematic.