You really asked two different questions, so let's take them one at a time:
Within the scope of 47 USC 230, Stack Exchange is defined as a provider of an "interactive computer service," which is defined as follows:
Interactive computer service
The term “interactive computer service” means any information service, system, or access software provider that provides or enables computer access by multiple users to a computer server, including specifically a service or system that provides access to the Internet and such systems operated or services offered by libraries or educational institutions.
Stack Exchange's users are defined as "information content providers":
Information content provider
The term “information content provider” means any person or entity that is responsible, in whole or in part, for the creation or development of information provided through the Internet or any other interactive computer service.
(Small caps removed to aid in legibility.)
Then, under paragraph (c) of that section:
(1) Treatment of publisher or speaker
No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider.
(2) Civil liability
No provider or user of an interactive computer service shall be held liable on account of—
(A) any action voluntarily taken in good faith to restrict access to or availability of material that the provider or user considers to be obscene, lewd, lascivious, filthy, excessively violent, harassing, or otherwise objectionable, whether or not such material is constitutionally protected; or
(B) any action taken to enable or make available to information content providers or others the technical means to restrict access to material described in paragraph (1).
Therefore, Stack Exchange is specifically and unambiguously excluded from being treated as the publisher of any user-generated content. Additionally, there's an explicit waiver of liability for moderation activities. This waiver does not contain any provision requiring Stack Exchange (or anyone else) to moderate "fairly" or to be "neutral," nor does it say anything about terms of service. Regardless, it's hard to see how they could violate these terms anyway:
Stack Overflow reserves the right to refuse, suspend or terminate your access to the public Network if it determines, in its sole discretion, that you have in any way violated these Public Network Terms or are otherwise ineligible to access or use the Network or Services. If your actions are determined by us to violate these Public Network Terms, Stack Overflow may, in its sole discretion, try to remediate that violation by working with you individually, but is under no obligation to do so, and if any such remediation efforts are unsuccessful (in Stack Overflow’s sole discretion), then Stack Overflow may revoke your rights to the Network.
A "public forum" in American Constitutional law normally refers to a situation where an agent of the government contracts with a private entity to set up a space in which the general public may speak or otherwise express themselves. Because the private entity acts on behalf of the government, the First Amendment is extended to the private entity, at least in the context of the space which it has set up. Stack Exchange, to the best of my knowledge, has no such relationship with either the federal or any state government. So the doctrine is inapplicable.
In California, the state constitution has a stronger provision which has been applied to private actors notwithstanding their lack of a relationship with the government. This case was Pruneyard v. Robins. In subsequent California caselaw, the courts have repeatedly refused to extend this logic beyond its original context (shopping malls with "common areas" where people may congregate), and it seems unlikely that it would apply to websites.