Can familiars do actions when they are summoned. If so how many actions and which ones?

All is explained in the rulebook, p. 17 :

When activated, a familiar may perform a move action following the same rules as heroes . Familiars treat any special terrain (other than obstacles) as water during their movement. The familiar may perform additional types of actions during its activation, as noted on its Familiar card . These additional actions, unless otherwise noted, may be performed in addition to the move action and may interrupt the move action in a way similar to hero move actions. If no such other actions are noted, then all the familiar can do on its turn is move up to its Speed value .

Familiars do not have anything like "Summoning Sickness", so they can be activated on the same turn that they are summoned. (However, because they weren't on the board BEFORE they were summoned, their activation must happen after the hero's activation on that hero player's turn).

Also, you can't activate a familiar, dismiss it, re-summon, and activate again. (It's still the same familiar and has already activated that turn.)

Familiars get 1 move action, period. Extras come from specific familiar rules. e.g. Reanimate also gets 1 attack action, so has 1 move and 1 attack total (no opening doors, reviving heroes, quest actions, double moves, double attacks, etc.)