If I use the recipe to make a crude weapon normal and than use the socket recipe how many sockets would a dimensional blade have? Would it be 1-3?

Yes, with the odds being skewed towards 3 (4/6 ~ 66.7%).

Horadric Cube socket odds may be found here.

Recipe to restore low quality to normal quality produces output ilvl=1. Hence the relevant case is M=3 (refer to the link below, subcolumn '< 26').
Dimensional Blade: https://www.theamazonbasin.com/wiki/index.php?title=Sword#Exceptional

Dimensional Blade with ilvl=1 should get Socketed (3) from Larzuk's quest reward.

1 Eld Rune + 1 Chipped Gem + Low Quality Weapon = Normal Quality Weapon of same type
ilvl = 1. That is, regardless of what the ilvl was beforehand, it will be set to 1.

1 to 3 sockets https://diablo2.diablowiki.net/Sockets#Sockets_by_Number

This post was edited by Marieusz on Nov 13 2018 04:35am