Is it possible to unlock the side door in the Temple of Hylia?

Solution 1:

As far as I know, it keeps locked while you're in the past. @Jeff Mercado seems to agree with me, and he seems to have gone through every pixel of this game. Furthermore, no walkthrough I could find mentioned any use to the side door.

As I see it, the door is there for consistency throughout the game, but does not serve an actual purpose in the past other than say "if there's a door in the present, then there has to be one in the past, unless someone went and altered the temple, made out of stone" at some point between the two times". Which brings us back to the path. Why would there not be a path on the other side? Why have a door that leads to nothing? This is of course speculation, but the path likely leads to Faron Woods and the Skyview Temple, as it does in the present. And since you don't need to access those locations in the past, the door remains locked. I suppose it also allowed the developers to not have to create a new, younger-looking location.

Solution 2:

If I remember correctly, that door unlocked after my first visit to the temple. It was the exit the Old Lady told me to go through to get to Faron Woods. It was my preferred entrance whenever I went to and from Skyloft (there is an easy to reach bird statue in a clearing behind the temple). Strange that it never opened for you.

Update: Confirmation from this walkthrough on Zelda Dungeon:

Run forward and speak with the old woman found here. After quite a bit of dialogue, she will unlock the adjacent door and tells you to open the nearby treasure chest. Run on over and open the chest to pick up some Revitalizing Potion. More importantly than the potion, this serves as the second bottle in your quest! With bottle in hand, head through the unlocked door.

Later in the game, you visit the temple in the past. During these visits, the side door remains shut and at no point are you allowed through it.