I agree, I don't understand what is reflexive about this particular exercise. The other response was not clear.
I think it might be the part about not having an object (a bike) that is reflexive, but it didn't follow the tips and advice (the -ся endings), unless it was one of the few exceptions that they mention they don't have time to discuss in the tips.
Either way, this felt like an easy and welcome break from the sheer volume of new vocab in this lesson :)
Yes, 'ве́лик', but it is common slang. 'Поеду покатаюсь на ве́лике'. Sometimes called simply 'вел', as slang too, but it is not used in all cases. We can say 'Заберу свой вел.' or 'Смотрю - нет моего вела!', and not 'Езжу на веле'. It may be 'bike' too, especially if bicycle is new and advanced, but more often 'bike' in Russian means motobike. The prefix 'velo' is used in compound words, as 'велодорожка' (track for bicycles), 'велопарковка'(parking for bicycles).
"I have no bicycle" would sound odd in most situations. I would say it is on the margin of being correct, but would discourage its use. A car salesman might say "I have no bicycles". But for one bike he would say "I don't have a bicycle" and is more likely to say with respect to the plural "I don't have any bicycles"
"I have no bicycle" is standard for me. A first search on Google for the exact phrase yields about 26,500 results.
The first suggested "correct solution" (I have not got a bicycle), which sounds really weird to me, yields 2 results. (See, for example, https://prezi.com/4ar1c8irvv7l/mi/ for one example.) Consequently, "I have not got a bicycle" is unacceptable (in my opinion). In contrast, "I haven't got a bicycle" yields about 104,000 results. The point here is that formalizing a common colloquial usage can create an absurdity (the "not got" grates on the ear of a native speaker).
"I ain't got no bicycle" is also common spoken colloquial English, but because it is regarded as uneducated, people rarely write it. Google yields about 9 results (nevertheless, significantly more than "I have not got a bicycle").
"I don't have a bicycle" yields about 334,000 results, while the more formal "I do not have a bicycle" yields about 389,000 results.