Google: voltage drop
It should pull up a chart of amperage vs length and tell you what size cabling you need.
Put the inverter somewhere where it can get plenty of airflow as it needs cooling.
If you hardwire, make sure there is a full disconnect (might be built in, if not add a seperate cutoff)
Battery bank needs to consider amp-hrs. If the TV draws 48watts and you run it for 3hrs in the evening, you need a battery bank that will allow you to draw 6amp-hrs (48*3/12). If you use other items, you need to run the same calculation (wattage*duration/12volts) and add them up. The battery you buy should have around 3 times the amp-hr rating as using the entire rated capacity will quickly damage it. (The coffee maker is an issue because while it is short duration, it's a lot of amps.)