As you know, I just moved into the Broken Oar Grotto which doesn't respawn anything.

Now my question is:

If I were to do something like put my blade on a table, would it disappear?

If it's in a house you own, i.e. Hjerim or Breezehome or Proudspire Manor, then no. Anywhere else, yes.

Edit: I just saw that you said Broken Oar Grotto. In this case, I have no idea. Sorry dude.

The reason I'm asking this is because the place doesn't reset.

Like you said, Broken Oar Grotto doesn't respawn. The storage is safe, you can leave your things and they won't disappear.

Okay, I just wasn't sure if the items that were left out of chests would stay.

I've been dropping ingot, ore, dwemer parts, etc. on the ground by the smelter in Whiterun for 25 levels and it's all still there.

I also dropped some Stormcloak armor in the Winterhold courtyard and it's still there.

There's probably safe places everywhere.....

Items dropped in-world don't despawn. That's why people with big save files get corrupt saves, from all of the dead enemy weapons being left around and cluttering memory.