yes the 500w inverter is MORE than big enough for TV, and charging cell phones etc
we don't know how big your solar panel is ?
we don't know how much tv you watch ? or how many lights you use etc..
all of this is more about the 'batteries' than the solar
yes the inverter connects to the 'house' batteries
just think of the solar as a low power battery charger
you might need some generator run time, for recharging, or spend one day a week with hookups
you run from the batteries, the solar does recharging during the day
how much battery do you have ? how old are they ?
