I emailed my sketch to a prototype company after signing an agreement. Is that agreement binding, and can I mail a picture of my sketch to myself and keep it sealed in case someone wants to patent and copy my idea?
The agreement is binding (to the extent of its wording and if they signed it to).
To protect your work from being patented by somebody else you can either file a patent/utility model/design or publish it. As the law grants the patent to the first person to file, its basically your problem if someone else patents this first. UNLESS it was stolen from you, in that case, if you can prove that, you can take over their patent/priority.
Mailing it to yourself will probably not help as they would claim that they invented it too, you need prove of giving it to them prior to their application.