Many questions. And some of them even have the same answer, (like the 1st 2)
Q1
oes a converter change 120V to 12v dc
A1:YES
Q2
oes a inverter change 12v to 120v?
A2: YES
C1:I have a Inverter that I plug into the cig lighter to run 120v things while dry camping.
C1Respnose: Those are handy for LOW power devices, may be enough to run a laptop, but your power limit is right close to 100 watts.
Q3: If you have a inverter, and it does change 12v to 120v, do some or all of the 120v outlets work?
A3: Depends 100% on how the inverter is installed,, IN my coach for example I have an after market Prosine 2.0 (2,000 watt, true sine wave, VERY NICE) and yes, SOME outlets work off the inverter,, Which ones depends on a switch I installed but here is a partial list.
Small inverters usually power just the Television/Rado/Related outlets.
Larger ones may add the GFCI controlled outlets (This includes bathroom, and your electric shaver)
over 1500 watt adds the Microwave.
I, added the outlets by the bed (Via a switch) this also adds the fridge in my rig, your rig WILL be different in this respect..
C2: We have a 08 Monarch and I am not sure if we have a inverter. I don't think we do.
Suggestion 1:
Pick up a pack of what are often called "path lights" these are standard 2-prong 120 volt plugs with Neon lamps (Quarter watt generally) built in them.. Like night lights.
Plug one into every outlet in the house.
With shore power all should light.. Pull the plug and if any remain lit.. YOU have answered most all your questions above. (And yes you have an inverter)