There ARE certain things that you can appeal after a plea, but it is not the plea itself.
In other words, if, for instance, pretrial you file a search and seizure motion (1538.5) alleging that evidence that was recovered was gathered in violation of the 4th amendment prescription against unreasonable searches/seizures, and that motion is denied and the defendant later pleads, the defendant can still appeal the judges ruling on the denial of the 1538.5 motion.
Does that make sense?
In answer to your first question, I don't believe there is anything that prevents a plea agreement from being in writing. However, it is not the norm. Most are done verbally "on the record" and there is nothing in writing other than the transcript from the court reporter and the minute order from the clerk.
There are things called "Tahl Waivers" which essentially are written forms that cover a defendant's rights and what the maximum penalty for the offense are. These are used in most misdemeanor courts and in some felony courts. A lot of the court will have the defense attorney write the terms of the plea agreement on the bottom of the front page, this is really an informal procedure, for instance on a DUI it would look like this:
36 months s/p, 390 +PA, AB541, rest., 90 day DLR.
(3 years informal probation, $390 fine + penalty assessment, 4 month alcohol program, restitution and a 90 day driver's license restriction).
However, the type of formal written plea agreements that you see used in Federal Court are not the norm in California State Court (at least in Los Angeles).