④ I picked this one because it's the means of transportation or in other words: by bike. This one was wrong. I don't understand the difference between のほう and でほう. That's the issue.
でほう is essentially nonexistent and ungrammatical.
When you want to say "X is more something", where X is a noun, it's Xのほうが~
自転車のほうが速い "(Going by) bicycle is faster."
スーパーのほうが安い "(Buying it at) the supermarket is cheaper."
If you explicitly state the verb, then the location is marked by で.
自転車で行ったほうが速い "Going by bicycle is faster."
スーパーで買ったほうが安い "Buying it at the supermarket is cheaper."
Do you see the difference here?
In the former, the noun is being connected directly to ほう.
For these examples, の is the only possibility.
で marks the location or means of an action and cannot be used to directly connect the nouns 自転車 or スーパー with the noun ほう.
On the other hand, 自転車で行く and スーパーで買う are marking the "means" 自転車 and the location スーパー as the means and location of the action verbs 行く and 買う. In these cases the verb takes the past (actually perfect) tense to modify ほう. These are similar to the "relative clauses" you've seen before. It's the entire phrase modifying ほう, not simply the noun+で.
⑤Not really sure either. Since it's a comparison it felt like the right thing to write. Not much of a good grammatical reason huh?
Your answer is correct, but it's not so arcane that the only way to puzzle it out is that it "just feels right."
The sentence very clearly is intended to means "(going by) motorbike is faster than (going by) bicycle" (i.e. "more so than (going by) bicycle, (going by) motorbike is faster".
Since 自転車 is the point of comparison (the thing that バイク is being described as faster than), より is really the only option.
(As Toritoribeさん said before, the point of reference isn't always explicitly stated. バイクのほうが速いですよ or 自転車より速いですよ are perfectly valid utterances to mean "It's faster to go by motorbike, you know." or "It's faster than going by bicycle, you know." Both of them are comparisons, but it doesn't mean you just use one particle or another because the whole sentence is making a comparison. The correct particle is dictated by what you're actually trying to say.)
edit for clarity