The California secretary of state will date the articles when they are received in its office, so even if you get them back 6 weeks later, the incorporation date and existence of the corporation will be the earlier date. However, if you are not using a service or lawyer to do it (which can get you evidence of incorporation within a day or so for an extra charge), you don’t know if any mailed in articles might have been rejected so you shouldn’t start taking action until you know for sure they are approved and filed. Wait till you have a copy of the articles in hand.
You want to get the money in the name of the business deposited in a separate bank account (to avoid commingling) and you won’t be able to start a bank account until you have the articles unless you have a really good relationship with a banker.
You want to be sure that your application code (copyright or other rights) and any other intellectual property are transferred into the corporation’s name after formation and that your money received is done properly with proper legal documents since it is the sale of a security (whether sale of stock or a loan in most cases) governed by federal and state securities laws.