Some Mythos Cards ("Sheldon Gang Turns to police for Aid!" for example) place monsters without gates on the board. Do these monsters count against the monster limit?

Oldster said:

Yes, they do.

Turambar said:

They do if the monsters are placed on the Arkham board. Some cards release monsters into the Dunwich or Kingsport street locations. In those cases, the monsters do not count against the limit.

That depends on the card. If it says monsters are "released" into the streets, then it would count against the monster limit and go to the outskirts if it was over the limit (As per the rules, Dunwich and Kingsport monsters never count against the limit, so I am only referring to the main Arkham board).

Certain cards, however, such as "the terrible experiment", have different rules. The terrible experiment specifically tells you to put 5 monsters on the card, that those monsters are put into play when the fail condition happens, and that players may choose whether or not to fight them when passing through Misk. U. streets. Such monsters are never considered on the board and do not count against the monster limit.

There was even an official errata on this :

Q: Do cards that affect monsters at Miskatonic University (such as the “Campus Security Increased” Mythos card or the “Flute of the Outer Gods” Unique Item) affect monsters spawned by “The Terrible Experiment”?

A: No. Those monsters are not actually on the board; they can only be fought as if they were.