If you daven more slowly than the chazzan, should you skip what someone who came late to davening would skip so that you can reach barchu with the tzibbur?
I have to agree with everything YDK said in his answer - except for his conclusion (which he stated first). My direct answer to the question is, instead, absolutely. The question is, which parts should you skip and when are you considered "caught up"?
You should skip parts of Pesukei DeZimra as outlined in the Shulhan 'Aruch (O"H 52:1), in order to catch up to the Tzibur. The point of this is to be caught up for Shemoneh 'Esreh. As outlined later (O"H 66:3), one should interrupt Shema' and its Berachoth to answer to Barechu.
Consequently, as a personal matter, as someone who is a) often able to keep up within a few Pesukim over the course of a handful of Perakim (especially Shema', since the Tzibur generally slows down there), but b) will fall behind by an insurmountable distance over a period as long (and generally fast) as Pesukei DeZimra, I believe it is appropriate and advisable to skip those parts of Pesukei DeZimra that are outlined in the Shu"'A if one is in danger of falling too far behind to be able to begin Shemoneh 'Esreh with the congregation. If you get ahead of the congregation and begin Birchoth Keriath Shema' on your own, you can and should answer Barechu wherever you are.
I am not making this up on my own, either. I have been advised that this is the correct approach by various rabbis over the years.
Absolutely not. Barchu can be answered at any time and if there were such an idea of skipping to say barchu in order, it surly would have been discussed regarding someone who came late.
However, this person has the same law as someone who came late and may skip in order to daven shemona esre together with the tzibbur. If he has time to skip and get ahead enough to daven sh"e with the tzibbur, he should do so regardless of when he says barchu.
I once heard that, according to the opinion of the ARIZAL, one should always go in order and recite all of the Davening (prayers in the Siddur), even if it will entail falling behind. In fact, in some Synagogues, the Chazzan will not move on until there everyone has finished reciting each section, and I think that it is a nice custom to follow because it is more fair for people who take a longer time to recite prayers so that they don't feel embarrassed.