Small claims appeal request dismissed by judge through ex-parte request based on untimeliness. What to do?

The facts stated in the question do not make sense. What do you mean when you say that May 6 "wasn't 30 days of May 4"? As for the supposed...

Can the Civil Case Information Statement be filed prior to the Appellant's Notice Designating Record on Appeal?

Yes. There is no order in which those two items must be filed, but the deadline for the record designation notice is sooner.

Does the order sustaining the demurrer have to be appealed as well as the judgment of dismissal on appeals (two fees?)

No. The judgment is what's appealable. The order sustaining the demurrer is not.

What Civil Code and/or Rule to I cite in asking for sanctions in the Court of Appeal?

Just in case you wanted an answer to your simple question instead of a lecture on why you need a lawyer: The applicable rule is California Rules of...

Was I supposed to file my Writ at the same time I filed the notice to file the writ?

I've never heard of a notice to file a writ. So far as I know, there is no such thing in California writ procedure.

How can the y dismiss a appeal on the basis they did when i did everything on my end ?

The question does not make sense. You say that you filed the appeal in October 2015; then you say that "the appeal was given to the file clerk" on...

After filing a Notice of Appeal sustaining a dismissal after demurrer, what are the immediate deadlines?

I assume you mean a notice of appeal from a dismissal after sustaining a demurrer. The first deadline will be to file a notice in the superior...

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Will i be able to persuade the appeal court to allow me to file my opening brief in any way. If yes, how?

(1) If the appeal has not yet been dismissed, you can submit the brief along with an application for relief from default. In that application, you...

If you objected (on the record) to what the judge did at the time, then you can raise the issue on appeal. If not, any issue is forfeited by the...

Is there a CA case that states a person can appeal a civil judgment because they feel the court did not abide by the law?

What a person "feels" or does not "feel" is irrelevant to whether or not the person can appeal a civil judgment. So you won't find a case saying...