[RPG] Can a Tiger get two attacks in a round and a possible third with pounce

Yes, for two reasons:

1. The last sentence is part of the description of the Pounce ability, implying it's an elaboration of what happens if the target fails the saving throw from the initial attack. That's most of the point of the Pounce ability; it's intentionally worded differently from the Mastiff's attack ability.

2. Even if that weren't the case and the last sentence stood alone, the requirements are still met:

   If [a] target is prone, the tiger can take a bonus action to make one bite attack against it.

   The target is prone, so you get a bonus attack.

No, the Giant Elk is given only one Action (and no Bonus Actions), and thus can only choose to do one of the available actions.

The text for Charge reads (MM, pp325)

If the elk moves at least 20 feet straight towards a target and then hits it with a ram attack...

Your Actions(and specifically those for Attacking) are as follows (MM, pp325)

1. Ram
2. Hooves

The Charge means you are choosing the Ram Attack.

The next round, if the creature is still prone, you may attack with the Hooves. Otherwise, it's just another Ram attack without the charge (unless you take the OA to move away and run again at them.)

And as Andras says, there ARE beasts like the Elephant (MM, pp322), that have viable Bonus Actions available to them.

Trampling Charge. If the elephant moves at least 20 feet straight toward a creature and then hits it with a gore attack on the same turn, that target must succeed on a DC 12 Strength saving throw or be knocked prone. If the target is prone, the elephant can make one stomp attack against it as a bonus action.

The (Giant) Elk, however, is not one of them.