I get it's a pretty basic question but I just don't have the information.

We know that many swords, knives, and spears pre- 1000 or so were pattern-welded by looking at historical pieces.

Is the same true for axes? If some were pattern-welded would only the edge be? Did it differ?

I have had a few roughly thousand year old Scandinavian fighting axe heads in my hands, but from the same find. They had a higher carbon piece wrapped around the edge, forge welded on, and quite noticeable even though covered with the classic thick black oxide typical of excavated pieces. Sorry, no photos; my clients don't like their stuff all over the internet. I wish they did, but not everyone is Wade Allen. :(

According to Vikes book about viking weapons, the axes were never pattern welded. At least in the viking age.
The norse word "mal" translates to "pattern",are believed to be for pattern welded weapons, such as:

malaspjot = pattern welded spear
malasax = pattern welded sax
And many names for pattern welded swords are known, blodida and blodrefill. Which are believed to be the description of the patterns of the fuller of the blade. The blood that runs down on the pattern.

There were an distinction of weapons for war and tools for everyday use. The axe was predominantly made as a tool and used for war at necessity. The sax was a longer bladed weapon in early viking age and therefore has a deeper meaning to its continued pattern welded construction. The pattern welded weapons was mostly produced from 700-950 in the viking age. From 950 to 1100 the pattern welded blades were less and less common. The hilts and sockets of swords and spears were highly decorative but blades were of more simple construction.
The daneaxe is a weapon of war but never pattern welded as far as i know, and it was manufactured from ca year 950.

As to your question of if the edges are pattern welded, if would say no they never were. Of all the pictures and x-rays of weapons from research and museums all over Norway i have not seen pattern welding done on the edges on any weapon or tool.