As far as I am aware you can choose the alternative providers anywhere in Greece. In the same way as you change provider in the UK, nothing actually happens to the physical supply. The different providers simply take over the billing. They buy power wholesale from the generator and then bill you for what you consume. Basically the new entrants are offering a discount to get market share and ΔΕΗ, the incumbent, is also offering a discount to protect its market dominance.
ΔΕΗ are offering 15 % discount provided you pay your bills on time. Greeks are notorious for paying late. Greece is the land of the post-dated cheque. Giving a 15 % discount puts them closer to the new entrants and tying it to paying on time encourages customers to behave better thus helping cash flow.
Protergia have the best deal, called Protergia Xtra, PROVIDED you have a landline/TV service through Cosmote. In that case they drop the standing charge and give 20 % off the cost of the consumed electricity, again dependent on paying on time. It is worth noting that the standing charge is not huge. I have a 3 phase supply and night-time cut price energy so mine is about as big as it gets at €20.40 per year. Most people will pay less than that. Also the deal is only guaranteed for 2 years so after that it may get closer to ΔΕΗ. The other Protergia deals keep the standing charge, apply the night-time saver rate to all electricity consumed for 6 months and then revert to 15 % discount. Thus it is only better than ΔΕΗ for 6 months.
ΗΡΩΝ have a couple of different deals. One offers a 10 % discount on the consumed electricity and keeps the standing charge but it also gives you two months electricity free when you sign up to them for 24 months. The other also gives you 10 % discount on the consumed electricity and drops the standing charge. Unless you are a very low consumer I would say these are not as good as the ΔΕΗ deal.
It is worth noting that these discounts only apply to the costs of electricity consumed. There are also charges on your bill to support the network grid, to pay for distribution and green charges that also depend on the amount of electricity consumed. These are mandatory and are the same for all suppliers.
Finally your council tax and the equivalent of the TV licence are also collected through your electricity bill.
It doesn't matter how you pay as long as you pay on time. You can pay in cash in a banks, post offices and some other outlets. You can set up a Direct Debit mandate or you can pay through internet banking. In the latter case it is worth paying it the day before it is due since the payment goes out overnight so, if you pay on the day, they might count it as received the next day, which would be late. There is a €0.30 charge for paying through Internet banking. I'm guessing it will be similar for a Direct Debit or for paying in cash.