I do not wash my eggs unless they have poo on them. No, you will not get chicks without a rooster and none will hatch no matter how long you leave them in the nesting boxes. The best thing to do is check for eggs every day to make sure they are fresh. If you need to, lightly write in pencil the date you collected them!
One of the many great things about this forum is that no one will laugh at your questions! We all had them when we began keeping chickens, no matter how long ago that might have been. It won't be long before you will be answering other people's questions, and reassuring them that the only stupid questions are the ones you fail to ask.
The only thing I would add to the great advice you've already been given is to be sure to use warm water when washing eggs, the shells are porous and if the washing water is cooler than the egg than you can actually drive the bacteria IN to the egg. I myself don't wash the clean eggs, and give the dirty ones a quick rinse & scrub in warm water with a plastic scrubbie.
You can help keep the eggs clean by keeping the nest boxes filled with clean straw as well as the space in front of them. Sometimes when it's rainy & the runs are muddy the eggs come in more dirty than at other times.