I then cast Doomed Traveler. Now, do I get to pay 1 in order to draw a card, because the printed power/toughness of the Doomed Traveler is 1, or do I not get to pay 1 in order to draw a card, because the effects count as the creature itself having a power of 3?
I'm leaning toward the latter, but I thought I'd run it past all of you just to be sure.
Thanks in advance for any help you can give me,
Because Doomed Traveler is not a token, it would be a 1/1 and you would get to draw a card.
But assuming you actually meant that Doomed Traveler had just died, giving you a Spirit token, the Mentor of the Meek's ability does not trigger. As long as you control two copies of Intangible Virtue, your Spirit token produced by Doomed Traveler has power 3, not 1.
603.6b Continuous effects that modify characteristics of a permanent do so the moment the permanent is on the battlefield (and not before then). The permanent is never on the battlefield with its unmodified characteristics. Continuous effects don't apply before the permanent is on the battlefield, however (see rule 603.6e).
In this case the mentor will trigger when the doomed traveler enters the battlefield because the doomed traveler is not a token.
However if it dies and the token spirit is put onto the battlefield the mentor will not trigger, because the intangible virtue's effect will always apply while it's on the battlefield. So when the token enters the battlefield it will enter as a 3/3, there will never be a time when it's not a 3/3.