Yea, everything you're describing is pointing a giant neon blinking sign right at the router that says "I'm the culprit!". The problem is something in DHCP - my bet is that the router got screwed up somehow and refuses to hand out more DHCP leases because either it has no more left in the range it's allowed to hand out, or else isn't actually capable of handing out more (ie: maybe if the router's DHCP server is limited to 5 DHCP addresses).
Try flushing all DHCP leases from the router - there might be an option to do that somewhere in the configuration for it. Failing that, just do a reset of the router, reset the configuration back to scratch.
That should fix your problems.
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