Is there a way to move the Inventory bar to either the left or the right of the screen? When I press "I" the Inventory appears right in the middle of the screen.

No, there's no way to move or resize any open panels. (Unless there's a mod for that, but I don't think there were ever too many mods made for this game.) Just get used to closing your inventory whenever you're not using it.
If you just want to keep an eye on how much room you've got left, I seem to remember there being a small numerical indicator somewhere along the bottom right-hand side of the screen, showing how many total empty slots remain in your pack(s), where the number will turn red when you're getting very low on free space.

That's too bad. I've noticed that Din's Legacy seems to have the same issue with the Inventory in the middle of the screen.

I'm admittedly not exactly super well versed in the various Diablo-descended PC-ARPGs, but I don't think more than one or two of those I've played have the inventory in a side panel and/or let you move the window around. (The original Torchlight does the former.) I can't imagine the inclusion of either of the above features became a very common thing in this subgenre until maybe the last four to six years.
It should also probably be noted that, as far as I know, all of Soldak's games -- going back some fourteen or fifteen years -- use the same (entirely homemade) game engine; the lack of moveable windows is probably something that the engine couldn't easily be made to support. (I've also heard that Din's Legacy is supposed to be the last hurrah for that engine. Hopefully this is true, and the next project will be on a different engine.) It could also be that this wasn't a feature the developer cared about, and he never got enough requests for it from players of the "pre-release" versions during the early access phases of his last two or three games.

Out of curiosity: what's the problem you have with this? I can't say it ever especially bothered me that the inventory was front and center, so to speak, especially given that I usually had the game paused whenever I was doing anything much with my inventory or any other personal/NPC window.

I just prefer to have the inventory either on the right or left side of the screen. When I'm moving inventory around, I want to see the screen in case my character gets under attack. With the inventory right in the middle of the screen, it just makes it more difficult to see.

In that case, you'll be happy to know that you can freely pause and unpause while you have most any of the these kinds of windows open. You can even pause the game in combat, give the order to use a spell or ability, then unpause and immediately have the thing activate (assuming you have enough mana, the ability's not on cooldown, you have the right kind of weapon equipped, etc, etc.), then pause again, switch weapons, drink a buff potion, "queue up" another spell, unpause... :) Obviously, neither health nor mana regenerate while paused, nor do cooldown timers run out; but neither do quest timers advance, nor do damage-over-time statuses that you're under inflict their effects on you, and so forth.
Basically, play it like it's FTL and pause liberally. ;P

Thanks! I didn't know the inventory still worked while paused.

Also, is the Demon War expansion worth it for Din's Curse?

It kind of depends on your tastes, to be honest.

Allow me to quote myself from a years-old post in another topic in this forum: Also check out this post and some of the immediately following ones in that thread for a little more info.

From another old post of mine that I think is also somewhat relevant to this question: