Both sentences are grammatical and make good sense. Neither is wrong. A person is certainly entitled to prefer one over the other in various contexts. Rather than thinking in terms of right or wrong, it is better to concentrate on understanding the nuances conveyed by the (grammatically correct) sentences. The first sentence uses infinitives, while the second uses present participles. The meaning of the sentences is exactly the same, but they "feel" different. Broadly speaking, infinitives excel at stating facts, whereas participles excel at creating images. Think about Shakespeare's famous line, "To be or not to be, that is the question". It creates a stark choice between two devastating alternatives. Had he said "being or not being, that is the question", it would have had the same meaning but it feels that way like a bowl of mushy porridge. On the other hand, think about the common saying "Seeing is believing". It creates in your mind an image of a person seeing and believing. Had the saying been "To see is to believe", it would mean the same but it sounds more like the result of a scientific research paper than a wise saying. As for the bicycle, the reason your teacher refers "coming" and "going" is that he/she likes to imagine pictures of bicyclists riding their bicycles. He sees the wind blowing through their hair and perhaps secretly would rather be riding a bicycle than teaching English at that moment. "Coming" and "going" gives him/her the images s/he likes and so prefers the present participle for that reason. With the infinitives, "to go" and "to come", you just don't see the wind blowing through their hair.

Your teacher is correct. You "spend (time) doING something, not "...to do" something. When certain verbs are followed by another verb, they are generally followed by either the "to infinitive" (I want to learn English), or the "-ing" form (I like learning English), while some verbs can be followed by either (I stopped to buy an ice cream, because it was so hot / I stopped buying ice creams because I was putting on so much weight). If you look in a good grammar book or web site, you can find lists of these verbs patterns.

Both versions are grammatically correct, but the second one is often preferred in everyday usage for its simplicity and clarity in describing the action of traveling

The infinitive is used to convey more general meanings. It takes me 30 minutes to go there. (This is general. It’s true now and would be true if you decided to go there in the future) I spend 30 minutes going there. (This describes your current practice) I spend 30 minutes to go there. (Unnatural. You spend time doing something) Stop spending your time watching TV. (Your recent activity) It’s more productive to read.