Go to your map and scroll down until you see teleport to base. You should also see, metropolis, gotham, event, and Washington D.C Doom
That’s the only way. Did you perhaps not do the base introduction? I don’t think you can hide the base from the map drop menu. Try relogging maybe that’ll help. More information is needed from you to figure this out.
That’s not the only way. In fact it is a “new” way. There was a time before the warp menu. The OG way is to go to your lair’s location in Gotham or Metropolis and enter it.
You cant. The safest way to protect your base from enemies finding the location is to not know it yourself.