If, after drawing one of the five cards from the face up train cards, and after replacing that card, the second card is drawn from the top of the deck and is a locomotive card, is the player allowed to keep that card? Likewise, if both cards are drawn from the the top of the stack and the second card is a locomotive card, is the player allowed to keep that card?
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It varies by which Ticket To Ride game you're playing.
In USA, Europe, and Asia, you may do any of the following on your turn:
In Switzerland and Nordic Countries:
The difference has a lot to do with how they are used - the Switzerland and Nordic rules don't allow engines to be used for plain routes, only for tunnels and ferries.
The general rule is that if a card is drawn without looking at it, then it doesn't matter whether it's a locomotive or not - you get to keep it regardless. So the answer to both your scenarios is "yes". In fact, if you're really lucky, it's possible to draw two locomotives from the stack.
When you are considering picking up a visible locomotive (i.e. one present in the five face up cards), then different rules apply. You can only pick a face up locomotive if it is the first and only card you pick up. So if you pick another card, either face up or from the stack, then you cannot pick up a face up locomotive.
Yes - just as if the first card drawn from the top of the (face-down) deck is a locomotive card, the player still gets to draw a second (which may also be a locomotive!). The key point here is that cards drawn from the deck rather than from the face-up cards are private; they're information that you don't have to share with other players, so in particular you don't need to let them know whether you've picked up a locomotive card or not.