What model is the router? If relatively new, you may have customer service available from Netgear for free. Their paid service, I'd be hesitant about. The one time I tried it, it cost me to have them tell me to reset everything back to default (which didn't help).
Because network addresses are assigned by the router, you’ll never be able to run a cable directly from the modem and use the router at the same time, as you noted. Also, the router provides Network Address Translation, which is a safety measure you’d best not do without.
It would be unusual for a router to restrict speeds unless specifically configured for that. Cables could be a problem. I’d suggest using decent cables -- not "audiophile" Ethernet cables -- that are Cat 5e, Cat 6, or Cat 7. Try swapping cables, as they do fail from time to time
After making changes, sometimes it helps to shut off modem and router and let them boot up again.
That’s about all I know. There is a Small Net Builder’s forum available online where you can get more free advice. Like the free advice anywhere (including mine), it might be good, bad, or indifferent.