Does Rand travel from the blight to Tarwins Gap when he takes the power from the eye of the world from Aignor?

Yes, but it is generally believed that this was before RJ had a really clear idea of what he wanted Traveling to be (as in later he realized that he wanted to be able to have armies travel also with the help of channelers). As such, in the prologue and in the end of tEoTW we see Lews Therin and Rand Travel without opening gateways.

I agree, it was a combination of Traveling and Skimming through the use of the stairs, and more of Rand being guided to take them by the ALL CAPS VOICE iirc.