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The two の's, what are their functions here?
If I removed any or both of them how would that affect the sentence?
Does では mean so then here, or is「のではどちらが」a canned expression?
一人で旅する(a)のと友達と旅する(b)のではどちらがいいですか？
Which do you prefer? Traveling alone or traveling with your friends?
新しいの
the new one.
新しいのが好き。
I like the new one.
の here acts as a "dummy noun" that is being described by 新しい. In your case, this dummy noun is being modified by the verb 旅する.
一人で旅する = to travel alone (verb)
一人で旅するの = travelling alone (noun phrase / gerund)
Likewise, 友達と旅するの is a noun phrase meaning "travelling with friends".
Your second and third questions both fall into one answer. The first reason why you cannot remove の from this pattern is because と can only list nouns (の is what allows 旅する to become a noun). The second reason is because this is a set pattern.
As for this grammar pattern, it is one of the main ways to create a list from which 「どれ、どちら、どっち」 can choose from. では here essentially means "out of" or "between". If this does not sound clear, here is an example:
猫と犬とウサギ（と）ではどれが好き？
Do you like cats, dogs, or rabbits the most?
More literal translation: Out of cats, dogs, and rabbits, which one do you like most?
Here is another example I stole from IMABI:
ジョギングは、朝と夜ではどっちがいいの
As for jogging, which is best, morning or night?
So for your example:
一人で旅するのと友達と旅するの（と）ではどちらが いいですか？
Literal Translation: Between travelling alone and travelling with friends, which one is better?
Natural translation: Do you prefer travelling alone or travelling with friends?
The addition of the second と makes it sound a bit more old fashioned. It is omitted often. In addition, と can be replaced with か when implying that you can choose from something outside of the list. You can find more information about this pattern here.
