In normal, day-to-day speech in Mainland China, you wouldn't hear anyone use 认为 in this type of question. Most people would ask, 你觉得....怎么样？
Yes, 觉得 is used more commonly than 认为, though such sentences can be used. 认为 sounds more book-ish and overly precise than 觉得, which has multiple translating uses.
Without 觉得 and 认为, we can simply express the targeted translation as
What are the differences between：
a) 觉得 vs 认为，
b) 表演 vs 表现？
for b) 表演 is more like a performance in terms of arts like a concert/dance while 表现 is performance in terms of work or result
你认为他的表现怎么样 should be accepted for the listening exercise. If you agree please upvote https://forum.duolingo.com/comment/29456371
认为 is more like 'to reckon'
I wonder if "What is your opinion about her performance?" could be accepted here. I reported it, just in case.
Are there any differences between 表演 and 表现 ? Duolingo uses both in this course