If I remember correctly, internal lighting is 12V bulbs. Thus, they run off your 12V battery - when shore power cord is disconnected from its source. To check, simply disconnect your TT's main power cord and turn a few lights on. If they work, then the light is 12V based (which runs directly off the battery).
To "size" your needed Power Invertor (from 12V to 110/120V), read the back of your TV and CD/DVD Player. View its wattage numbers and "double check" on their manufacture Web sites as well. For example, an LCD TV might use between 90 - 130 Watts - as described within: http://wiki.answers.com/Q/How_many_w...nch_lcd_tv_use
After obtaining the TV's wattage usage, do the same wattage investigation with the DVD/CD Player. Perhaps it consumes 75 Watts. Add both together and one gets their "on average" combined wattage size. For example, total of 100 + 75 (as examples).
Do the same "wattage" check for items plugged into the wall outlets. Remember that coffee pots and hair dryers love watts.
The Power Inverter connected to 12V battery doesn't power the Wall outlets. Only its own outlets are live. Thus, your usual TT 110V devices would have to be unplugged from the TT wall outlet and plugged into the Power Inverter's outlets. Plugging the TT's main power cable into the TT's shore power cable would create a loop. Thus, best to have Power Inverter connected directly to 12V battery - using thick 10 - 8 guage wiring. Higher the Power Invertor wattage load, thicker the 12 wiring (to its 12V battery source).
Hope this helps in your research....