I'm a sub-contractor for a dog walking business. I will be issued 1099s and be responsible for my own taxes. Am I able to pay my taxes quarterly? Am I allowed to write-off everything I use for work (my car, my cell phone, my computer)?

Yes, you are liable for all your own taxes including what they call "self employment" taxes. And yes you can pay estimated quarterly taxes.
And you can write off.

If you have a phone and computer that is used only for business, then they can be deducted. You will need to keep a milage log on your car, and separate personal and business miles.