This is possible although it would make more sense to connect the pi to the router, then everything else to the pi -- I mention this because the "along with everything else" in your question is a little ambiguous. Perhaps that is what you meant. Anyway, this would obviously require a bit more hardware -- a switch or hub and/or an appropriate wifi adapter.
There are advantages and disadvantages to this. The major advantage would be you could set up a much more fine tuned firewall than you can with the stock software on most home routers. But the disadvantage is...
I am guessing that if so it would limit all traffic in the network to the throughput the pi is able to output (and therefore in most cases not very useful), is that correct?
It is. If you want to support other people streaming on multiple devices simultaneously, the pi will be a potential bottleneck because of the max 100 Mbps ethernet speed. That would not exactly have to be divided since output would be on the USB bus -- "not exactly" since the pi's ethernet is already on the USB bus, which is in theory max 480 Mbps; so when the ethernet is maxed out, you still have enough left to pass though on the USB bus. Incoming vs. outgoing is not such a big deal since normal internet usage is mostly in one direction (in).
I think you want at least 5-10 Mbps available per device and leave plenty of wiggle room in the numbers; so half a dozen might be realistic. If those devices are often used for torrenting or otherwise mass downloading, forget it. The people using them won't be happy.
I am sure the processor + iptables (the linux kernel firewall, which is also used in most routers, but hidden behind a simplified interface) can handle that volume, so that is not a problem. But if you are seriously considering this you'd want to examine what the usage is on your LAN.