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Would you just let DHCP assign all the IPs on this network pretty much...........?
Probably not... I'd use a mix of Static addresses and DHCP. For the workstations, DHCP is fine, and recommended. For the servers, printers, and various other devices, I'd do a static IP address.
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Also, would you be ok with cat5 cables or would you benefit much from cat5e/6/6e or fibre etc....?
Where possible, I'd use cat6, especially if I'm building the site from scratch.
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Also what if your router died how would you have a back up router (or switch if that died) so that you never lost connectivity or can you do this?
Sometimes you have to plan for that and have a backup plan in place. If 0 downtime is allowed, then you'll have to pay for that and keep a backup router and/or switch on hand in case of a failure.