Unless it is important for the context, I would say it is absolutelz unnecessary to translate. Duolingo is kind of "gender neutral" and accepts both the male and female forms of German words and don't except you to mention the gender in the English translation. Just examples to show what I mean:
- Ich bin auf der Suche nach meiner alten Lehrerin (no need to mention if it is a female, I would say, it is not relevant, unless it would be in a bigger context that would somehow require it)
- Sie war meine Mathelehrerin, eine ziemlich junge Frau. (translating it to "female teacher" would be plain weird, because the other parts make it very clear)
- Es kann den jungen Lehrerinnen ganz schön schwer sein, wenn die Jungs in die Pubertät kommen. (Here it is obvious we are talking about only female teachers, so it would have it's place)
to search is totally okay translation to my ears. Absuchen however would sound weird in this context. "Ich habe das ganze Gebäude nach der Lehrerin abgesucht" would, according to Duden, be okay, but "Ich suche die Lehrerin ab" would probably mean that you are searching the pockets and the bag of the teacher, that she has on her.
absuchen is a bit like durchsuchen, but less frequently used. Duden says that one meaning of absuchen is "gründlich durchsuchen" :)
Without the "for", the sentence takes on a meaning that is particularly relevant in law enforcement contexts: one would be "patting down" the old teacher, checking her pockets and whatnot for weapons or contraband.
Similarly, if I were to "search your house", I would be looking through closets and cabinets, under beds, etc., generally trying to find something inside of your house. On the other hand, if I were to "search
for your house", I would be going down streets, trying to locate the dwelling itself, so that, perhaps, I could then search it.
It is somewhat ambiguous whether "eine Lehrerin/Frau/Tür" is "one lady teacher/woman/door" or "
a lady teacher/woman/door". Usually the "eine" would be italicized if it were to mean "one" ("Ich suche eine alte Lehrerin.") and otherwise should be considered to be "a", which is what we have here.