For service by US mail, they will attach a "proof of service" to the mailed document that contains a declaration by the person who placed it in the mail. (I think other jurisdictions may refer to this as an 'affidavit of mailing.') The e-mail notice is an informal preliminary, not the actual proof of service.
As to the prohibition on party service, it's to discourage fraud and avoid direct confrontation. It's important to comply because if you don't the court lacks personal jurisdiction and any judgment or order issued is void.
Here's a case on it:
Caldwell v. Coppola
219 Cal. App. 3d 859
Court of Appeal, 4th Appellate Dist., 1st Div. 1990
The first Practice Act limited personal delivery of the summons to the sheriff of the county where the defendant may be found...Although the Practice Act was amended to permit private individuals to serve notice, the common law rule consistently prohibited an interested person from personal service on the opposing party...In 1872 the Legislature enacted section 410 limiting personal service to a nonparty or the sheriff of the county where the defendant is found...In prohibiting personal service of process by parties, the current section 414.10 continues the intent of section 410.
The long-standing prohibition on personal service by the opposing party arises from the adversarial interest present in legal actions and the concern for discouraging fraudulent service. "The common law rule was that an interested party could not serve a summons, the policy behind the rule being that an interested party should not be put in a position whereby he might gain an advantage over his antagonist." (Com. (1929) 3 So.Cal.L.Rev. 129.)
Although attorneys are competent to serve process, the prohibition on service by the opposing party is strictly enforced. (See Sheehan v. All Persons(1926) 80 Cal. App. 393 [252 P. 337].) When a party has served notice on the opposing party, the court lacks personal jurisdiction over the defendant. (In re Marriage of Smith (1982) 135 Cal. App.3d 543 [185 Cal. Rptr. 411].) Personal service by a party renders any judgment or order arising from the proceeding void, despite the defendant's actual notice. (Sullivan v. Sullivan (1967) 256 Cal. App.2d 301 [64 Cal. Rptr. 82].)