I guess the first thing to visit is what ONT you have installed at your house. Older ONTs are limited to 75 mbps regardless of coax or ethernet wiring.
The old black and red Actiontec router is 2.4 GHz wireless N. Depending on the environment, that router probably won't get over 50 mbps. You may just need a better router or wireless access point that is dual band 2.4/5 GHz wireless AC or AX(wifi 6). To put things in perspective, you only need 25 mbps to stream 4k and 8 tk stream 1080 which is the majority of content.
To answer your questions...
1. Yes. Older ONTs are limited to 75 mbps on coax or ethernet. Newer ONTs are limited to 100 mbps on coax.
2. Yes. Since you are dropping TV you can get a pair of MoCA 2.0 adapters. Setup would be...
ONT > ethernet > 1st adapter > coax > 2nd adapter > ethernet to router.
3. Mesh routers are very popular, but any router with gigabit ethernet ports and dual band wireless AC or AX will work.
4. If you have an older ONT then yes 100% a tech needs to come out to upgrade the ONT. If already a gigabit capable ONT you can run ethernet yourself and call Verizon to switch to ethernet. Then you'll be able to upgrade to any speed Verizon offers.