Power adaptors are readily available that plug into Australia's 240v outlets, converting to 120v. Find them in travel, luggage and electronics stores.
120v or 240v. 120v is one leg of the main panel, and 240 is two legs of the main panel. 120v is lights,outlets. 240v, dryer,stove.
No. The neon sign is fed by a step-up transformer. Primary side 120V, secondary side 7500V. If you applied 240 to the primary side you would get 15000 volts on the neon tube. A flash over and then nothing. If you can find a transformer from 120V to 240V or 240V to 120V then you are good to go. Connect 240V to 240V side and you will get 120V out the other, connect the 120V side to the neon sign and you should have light. Transformer should be at least 100va. This will give you an output of .83 amps at 120V
120V takes less energy
You cannot use 2-120v outlets to power a 240v dryer. You can convert a 240v dryer outlet to power 2-120v outlets if they are supplied with a neutral. This requires a competent electrician. Do not do this yourself.
Because they are "in-phase". In order to get 240v, you need two 120v Alternating Current lines that are 180° out of phase, that is, opposite phases. Only when one line is +120v and the other -120v will you see 240v between the wires.
In the US, both 120v and 240v will be needed for your home, as different appliances need different voltages. Your TV needs 120V, while your electric dryer and stove will need a 120V/240V supply. If you have an electric water heater, or central AC unit, they will need a 240V supply.
you get an adapter
Off hand no but your explanations are not clear as to who is doing what to whom,,