Does anyone know of a reliable company that specializes in monthly apartment rentals in Seville?
This isn't Seville-specific, and it includes private apartment owners who rent out their place if you're looking strictly for a company that rents out the properties, but in Madrid 7 months ago we got a great place thru HomeAway (www.homeaway.com). Their sister Website is VRBO (www.vrbo.com).
If you're staying for 2 months and want to get around like a lot of locals, plan on getting a scooter and cutting in and out of traffic ;-)
Those sound like good places to start. The city center is quite walkable, but if you choose a place further out, then check public transit options at www.tussam.es. Myself, I'm afraid of scooters ;).