There are 3 things I believe affect the quality of the work. The skill level in Hideworking is key, the quality of the knife is important, and fatigue, either from weight carried or exertion.
For best results, get Hideworking skill up. My grandmasters routinely work skins while traveling, using just a tree and a rock, and maintain Fine quality furs. This, even when I don't strip down to reduce the burden malus. The fastest way to improve the skill is by trapping the plentiful birds in the Unreal World.
Knife quality helps to make up for lower Hideworking skill but, by itself, it's not the decisive factor, just the leading factor. And, if the hide is of fine quality, or better, don't work it until fully rested. 'Lively' is NOT good enough. Get to 'Vigorous' before you start and improve the odds.
I doubt that table vs tree trunk is a factor because the game doesn't quantify the table's value. It doesn't care if you used Perfect boards or inferior boards. It's still just a table.
I think the beater quality may make a difference but there's no way I can prove that. I always keep making clubs until I get a Fine club. Then I trade for the first GrandMasterwork Mace I see and use that as the beater.
I've seen Grandmasterwork furs as treasure. I've never made one, nor skinned one at that level. It may be impossible for the player to achieve such results, despite arranging all the factors to maximum advantage. It's indicative in that Superior skins often don't get finished successfully, despite maximized conditions and factors.