Should do yeah, crossover is what it should be but if either the router or switch supports auto-uplink then it won't matter, patch will work fine.
I assume you are referring to the Linksys looking device as a router ("modem/wireless/switch") - yes it should show up as another node on the network. It will need to include some form of routing for the other PCs to be able to access the internet through it, I assume you are implying it is a router.
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