When the app asks you to search for a mission but you have not completed the last one can you begin a second mission?

No, you can only have one regular mission active at a time. when the App tells you to trigger a mission and you already have one, you are allowed to look at the top two missions and decide if you want to keep the current mission and discard both newly drawn missions or to replace the current mission with one of them, then discard the old mission and the second mission you drew.

When the app asks you to search for a mission but you have not completed the last one can you begin a second mission?

You always draw the top two missions, but you can choose whether to replace the existing mission or discard both the drawn missions.

If you replace the existing mission, then you discard any success tokens or undefeated aliens, and return soldiers to their reserves; otherwise anything already on the mission stays there.