In Arkham Horror, if I lose a turn and then am drawn into a gate, am I then only delayed?
Losing your turn is different to being delayed. Being delayed means you cannot do anything during your movement phase
If you have lost your turn, you skip all phases (including upkeep) except for the Mythos phase.
From the FAQ:

Q: If a card tells you to “lose your next turn,” does this mean skip
all phases, including Upkeep?
A: Yes, all phases except the Mythos
Phase. Note that this is different from being told to “stay here next
turn,” which means you are delayed.

For your original question, based on the above, I would say you can only get sucked in to the gate that appears as part of the Mythos phase and then you have both lost a turn AND are delayed.
On your next turn, you cannot stand as that normally happens in the movement phase. You also do not have any encounters. The turn after that, you get to stand and have encounters. The following turn will be when you can actually move. This way, you still have at least 2 encounters in the other world (unless you get lucky - but to be in this situation, you need to be really unlucky)