Hallo RobS49 und sfpugh,
I can totally understand - German prepositions can be tricky.
In regards to your first question, RobS49, the literal translation of the sentence would be "The young man rides on the new bike." The reason they use auf here is because you would normally sit on a bike - "auf einem Fahrrad".
Nevertheless, it would also be correct to say "Der junge Mann fährt mit dem neuen Fahrrad." - "The young man rides the new bike."
Sfpugh has also given a very good answer to your second question. Aufpassen is a separable verb which means it contains a separable prefix (auf in this case). When you use separable verbs in a sentence, the main verb goes in its normal place - the second position - but the prefix goes last. However, you still need the preposition auf in the sentence as well.
If you would like to learn more about seperable and inseperable verbs you can have a look at Module 12 lesson 12.5 and 12.6 Inseparable Verbs (Part 1 and 2).
The preposition auf (as well as über, unter, vor, zwischen, an, neben, hinter and in) can either be followed by the accusative or the dative. You can have a look at lesson 4.9 Prepositions with the Accusative for more information.
I hope this helps! I will still pass your feedback on to our development team to look at.