Legal Question in Civil Litigation in California

It is nothing to do with the State Bar, although doing that might run afoul of more general Rules of Professional Conduct that could lead to discipline by the Bar. There are also plenty of lawyers who associate or affiliate with firms for particular work, while maintaining their own practice as well. I personally do that. I have an ongoing relationship with a San Jose firm that I assist on a case by case basis, or handle a particular transaction for them. I also have my own clients who have no connection to that firm, and I have a mediation and arbitration practice where I act as the mediator or arbitrator, again with no connection to that firm. As long as it is disclosed as required (or in some cases NOT disclosed, such as the ADR work I do) to the firm(s), and there is no employment or contract agreement prohibiting it, there is no problem. The only problem is if the lawyer is either a full blown employee of the firm, in which case they are obligated to devote their full attention in the practice of law to the firm, or there is a contractual limitation on outside work.

There's no law against it. Each firm can set its own policy. Lawyers who work full-time for a firm probably need its express permission to do other legal work on the side, since the employer is paying for all of their legal work. Some firms will give permission more readily than others, and most will want to know who the clients are, who's on the other side, and what the work will entail. Lawyers who work part-time probably don't need express permission, but they still must abide by their firm's rules.

Lawyers who hold side jobs must still obey the same laws and regulations as other lawyers, including those concerning conflicts of interest. That's usually feasible, though it's not always easy.