I believe what you're doing is known as click fraud, and you're probably seeing their efforts to stop this from happening. Companies don't like it when people do what you're doing, as it doesn't benefit them. There may be specific laws against this, but if all else fails they can probably get you under the Computer Misuse Act (roughly paraphrased: you're not allowed to even look at our computers unless we say so) (UK-only, other countries probably have equivalent laws). So you might want to rethink your automated clicking.

Anyway, a proxy is a special server designed to forward web requests. Proxy servers can also be configured to cache request, to reduce bandwidth usage. These tend to be used as part of coporate LAN systems to reduce the bandwidth used, control web access, and log visited sites. You also commonly see them in schools and universities - my secondary school used Censornet (a combined proxy, web cache and filtering system), and my current university uses a proxy server to reduce bandwidth usage and restrict internet access to authorised users. A firewall is not a proxy server, nor is a router, NAT.

Your Linksys router does NAT (network address translation, also known as IP masquerading, it's the technology that lets multiple computers appear as one to the outside world), and has a hardware firewall designed to prevent most attacks. Being a hardware firewall, it should not affect any of the programs on your system (hardware firewalls block on a per-port basis, not a per-application one). The router does not have a proxy server. Your ISP may have one, but the fact that your clicking program is working at all implies that it is correctly configured for whatever system your ISP uses. So I think we can safely rule out proxy problems, which leaves a buggy program or unhappy advertisers. I'd guess the latter.