[RPG] How many Opportunity Attacks can a PC make during a round

Yes. Say for instance a creature in melee casts a spell that requires a ranged touch attack, it would provoke two attacks of opportunity, one for the casting of the spell and one for the ranged attack in melee.

You still can make only one attack of opportunity for each opportunity that your foe gives you. For example, if you have Combat Reflexes and a Dexterity score of 15 you can make up the three attacks of opportunity each turn. You could make all three of them against the same foe, provided that the foe does three different things that provoke attacks of opportunity. (Link)

While the example from the article doesn't specify the situation I've outlined above it clearly states that though each individual opportunity does not provoke more than once, you can make multiple attacks against a creature provided they offer multiple opportunities.

No, because the Action is not used to move

If you had to use your Action to make the carpet fly, then you definitely would provoke an opportunity attack, because you only avoid that when:

you teleport or when someone or something moves you without using your movement, action, or reaction. (PHB, p. 195)

However, the consensus is that a carpet of flying does not require your Action to move, but rather to activate. The spoken directions you deliver to it are not an action, but something you can do in your turn without spending any resources since:

Your turn can include a variety of flourishes that require neither your action nor your move.

You can communicate however you are able, through brief utterances and gestures, as you take your turn. (PHB, p. 190)

The Carpet Doesn't Provoke Opportunity Attacks Either

If you considered a carpet's rider to be "mounted" on the carpet, one might be tempted to make the argument that the carpet itself provokes an Opportunity Attack, and the rider would then be attackable as well since:

if the mount provokes an opportunity attack while you’re on it, the attacker can target you or the mount. (PHB, p. 198)

However, a carpet of flying is an item, not a creature. And the rules state:

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. (PHB, p. 195, bold added)

Thus the carpet does not provoke opportunity attacks itself.