"I like that red dress."
Translation:A mí me gusta ese vestido rojo.
The me is required for I like. The a mí is not necessary, and the sentence should be accepted without it. It's used to emphasize the I, and there is no context to require its use here.
You may know by now since it's been 4 months, but if not, I look at that as an emphasis that Spanish uses. Now I just interpret "a me mi gusta" like this: "as for me, I like..." or "a ti, te gusta" or "as for you, you like", etc.
See above although A mi me gusta... was given as an alternate correct answer.
Verbs are conjugated for time and person, not to match the gender of the noun.
Reflexive (or defective) verbs are conjugated (mostly) in the third person (plural and singular) because the subject of the verb (here) is 'vestido' not 'me'. You can see this in the 'tips' for this skill: https://www.duolingo.com/skill/es/Recreation/tips
This is a complicated subject and spending some time understanding the structure of the use of these verb types would be worthwhile before moving on. Caution: grammar skills (sentence structure, direct objects, and reflexive objects as well as the use of the personal A) are needed.