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2. My router is an old WR740N, would it work correctly with a proxy server? Performance-wise, say I download with full speed and also use the proxy, would the router be able to handle it, or just heat up and die? Do I need a lot of resources/performance for proxy server?
Routers don't "heat up and die" except in exceptional circumstances, you'll just get reduced performance. The TL-WR740N can be flashed with the BREED bootloader, which supports altering the processor clocks; you could underclock them to sacrifice speed in exchange for less heat, and/or open the router and put a small heatsink on the CPU or SoC.
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3. I would also like to use my home proxy at work, to access prohibited sites, will it work externally?
Depends on how good your workplace's filter is. In a worst-case scenario you may have to run your proxy on port 443 with stunnel.