Once on Santorini you can either drive or take a bus from Fira or the port.

Car, motorbike or bus

Didn’t see any when there.... but then not large numbers of people due to Covid and some places closed for the summer

Caution should be advised everywhere but I don't expect you would have a problem. I left tablets/mobile and money but I was close by. Enjoy.

Either will suffice - the sand is quite hot so as long as you have something on your feet while walking you should be alright. Cabs are expensive on the island and try to rip you off. I would suggest either renting a car from the airport or using the local buses - which are cheap, but less reliable and come once every 20-30 minutes but go everywhere on the island.

Early October should still be warm and sunny. Towards the end of the month getting cooler

I’m not sure. There’s the rule “lunch here and free sun beds”. My girlfriend and I were in one of the southern beach and they explicit told us “First or second line sun beds free if you eat here. If you think you’ll go away in two or three hours, you can take the last line”. But I’m not sure everyone use this rules.

Appena si arriva sulla spiaggia, ci sono dei cartelli fuori di ristoranti che comunicano questo servizio. C'è l'imbarazzo della scelta: vedete cosa vi piace e poi scegliete!

Ml I went in May and the black sand was too hot to manage even with flipflops, but it looked lovely for swimming MN

Take your trunks! The sea will be warming up at that time but it will not be at it's warmest that happens a little later int he year. Whatever you do enjoy yourselves - a lovely island but watch out for the crowds in Oia at sunset.