Wow, there are a lot of questions there. I'll try to take them one at a time.

It is my understanding that Columbus significantly underestimated the diameter of the earth and this was one reason he thought he could reach east Asia.

Yes, Columbus appears to have thought that the earth's cicrumfrence was about 25% smaller than it actually is. We know that he corresponded with Paolo dal Pozzo Toscanelli, a Florentine scholar who believed that sailing west would be a feasible way to reach Asia. Toscanelli had a map, possibly based on Martin Behaim's globe, that showed Asia extending much further east than it does, and the Azores, Canaries and Cape Verde islands extending much further west than they do (the Azores would end up right off the coast of Labrador in Behaim's projection).

Still, Toscanelli thought the Atlantic (from Asia to Europe) was about 5,000 miles wide; Columbus on the other hand interpreted Asia as being much bigger, thinking the Atlantic was only about 3,200 miles wide.

However, he also would have had to have some estimate for the "width" of eurasia. How accurate was he on the "width" of Eurasia and where did he get his information from? Ptolemy?

Yes, Columbus seems to have been influenced both by Toscanelli and also Behaim, and by a general contemporary thought that the Indies (that is, what we'd now call Indonesia) were a larger island chain going eastward, and that there were even more islands to the east of that -- particularly, it was thought that Japan was larger than it is, further east, and on the equator.

Ptolemy did estimate that Eurasia spanned 180 degrees of latitude (it's closer to 130).

Weren't longitudes only reliably calculated in the 18th century?

The "discovery of the longitude" is dated to about 1760, when a method using lunar observation (either of the earth's moon, or the moons of Jupiter) to calculate time at a fixed point and thus the distance from the mariner to that fixed point, thus finding longitude. Harrison's H4 marine chronometer was built in 1761, but chronometers did not become widespread until the early- to mid-19th century.

EDIT: I a word