There are some gaps in the floor near the door frame. I am a new homeowner that's why I am not sure whether it is by design or it needs to be fixed. It looks ugly, can I put some kind of filler to hide the gap?
I am also noticing some gaps in the floor tile, what kind of filler can I use to fill them?
You've shown us two different things. The first is poor workpersonship. Jambs should be undercut to allow flooring to slide underneath, and flooring planks should extend further under transitions.
There's no great fix for this other than reinstallation. Fillers will probably won't hold up well due to movement inherent in floating floors, though they may be a slight improvement. If anything, use color-matched pure silicone or urethane caulk. They're flexible and won't shrink much during cure.
Trim overlays would just look odd there. You could look for a wider transition strip to replace the existing one.
In the second case we see seasonal shrinkage, or shrinkage due to the failure to acclimate flooring to its install location before installation. Chances are those gaps will close up as temperature and humidity rise. If not, you'll need to look into ways to fill and refinish the flooring, or ignore it. It's mostly an aesthetic concern and hard to avoid with solid wood flooring. Really, it's not supposed to look like the machine-printed facsimile in the next room.
