I'm connected through a LinkSys router with a custom linux on it. Is there any way to see if this router logs my network traffic, reading headers or something else? I don't have access to the router, though.
If it isn't manipulating your packets (beyond NAT), then it is quite impossible for you to tell if your traffic is merely being tracked based on looking at packets on either side of the router.
The smoking gun, as it were, would be to catch traffic that looks like a log upload. To do that you need to monitor the traffic on the outside of the router, have knowledge of what kind of traffic was happening on the inside of the router, and compare the two for TCP conversations that don't exist on both sides.
If your traffic is traversing a physical piece of hardware then it is possible for that device to capture and/or manipulate your traffic. Encryption addresses some of these concerns and allows for data integrity end to end. The device may be logging your traffic, but without access to the box, you can't tell what its doing. I suppose a clever person could put a sniffer on both the inside and outside interfaces and see if the traffic has been manipulated (NAT is of course going to manipulate it).
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Honestly, there probably isn't, other than trying to visit some unsavory site and seeing if whoever controls the router mentions anything. If you are worried about your connection being monitored, consider tunneling your traffic through SSH to an external server, or looking for a cheap VPN service if you don't have one.