Division commanders appear to be in a odd state of limbo to me. Unlike their subordinate officers, it seems like they would have responsibility over far too many troops to make useful decisions on a tactical level, and yet it also seems like they have too few troops to make a decision on the strategic level. What were the duties of a division commander? What was their average day like while in combat?

To say that a division commander can't make useful decisions at either a tactical or strategic level would be selling it a bit short.

Keep in mind that the structure in the military - both in WW2 or today - is designed with increasing levels of responsibility.

So sure, a division commander in WW2 might not be leading a platoon of men to take out an individual defense, but he's in charge of all the platoons, their companies, and their regiments underneath him in charge of taking the entire sector of a defensive line.

And that also means the logistics behind that (how you get the ammunition, food, repair parts, etc.), the coordination with other forces supporting said operation (be it air or naval forces), and even what seems mundane like setting the SOP (standard operating procedures) which guides how the unit acts.

Patton, as written in his 2006 biography Patton: A Biography by Alan Axelrod, implemented major reforms to II Corps after taking over following his predecessor, Lloyd Fredenhall, being relieved after the Battle of the Kasserine Pass. He implemented sweeping changes to adhere to strict military protocol and discipline that was felt throughout his chain of command.

Speaking of chain of command: division commanders are part of the delegation of responsibility that starts from the top.

Sure, General Eisenhower was Supreme Allied Commander and may decide what he wants the Allied forces to do - but he will delegate the details to his subordinates.

Ultimately, his staff will figure out which unit is most ready/available/capable of achieving the next step in objectives, and that order ultimately gets passed down to his Army commanders, their Corps commanders, and then their Division commanders who are responsible for their own set of objectives and how to get that accomplished.

An example covered in this would be that of Major General James M. Gavin, who commanded the 82nd Airborne Division during World War II - while "just" a division commander, he was certainly instrumental in shaping their capabilities, doctrine, and tactical planning to support bigger strategic goals. The Sword of St. Michael: The 82nd Airborne Division in World War II published in 2011 covers a lot of this - orders from above would have told him to take a certain set of objectives on D-Day, for instance, but the details of which parts of his division would drop where would be up to him.

That certainly isn't a small responsibility nor unimportant.

I would suggest it's better understood that it's predominantly dependent on the doctrine of both the military those field commanders serve and the individual they are immediately subordinate too. For instance a micromanaging Corps or Army commander might not give his divisional commanders a lot of responsibility as they delight in laying out every detail and expecting it to be followed to the letter of law with no leeway. Other field commanders might be substantially more free flowing allowing for those divisional commanders to seize independent initiative and to some extent independent command as they are the man on the ground. Germany is a fairly good example of a WWII military that leaned more towards this "man on the ground philosophy". The British were more towards rigidity of command with both philosophies having their pros and cons and each individual commander exercising their own prerogative in how extreme they took this philosophy. Rommel was an extreme version of this German philosophy and Montgomery was, perhaps extreme is too strong, but a tight adherent of the more rigid British philosophy on command. Suffice to say that much of this is contextual and all philosophies evolve during the course of the war.

Your second point again strikes at the heart of the philosophy both militarily and individually. Divisional commanders duties were in the maintenance, alignment, and readiness of his own command and to work in conjunction with the other units in his command. The necessity of divisional commanders stems primarily from the need for this delineation, though I would suggest it's more of a blurring line, of tactical and strategic responsibility. A Corps commander or worse Army commander should not need to seize individual regiments or brigades during the course of a battle and place them according to the needs of combat. This is not only time consuming and tedious it is also dangerous as it narrows the focus of these commanders to each individual crisis as it appears. The ability of a divisional commander to manage the crisis as it comes on their level alleviates and dilutes the burden of command. They can make both tactical and strategic decisions rather than being in a limbo they are often the backbone of any successfully functioning army. Their responsibilities aren't simply important but paramount to the overall success of an army because a Corps or Army commander simply cannot be in every place at once and he needs talented individuals across his command to meet the big and small problems as and when they come.