How do I find out what sealant was used before?
The answer to that is as you'd guess... look for product packaging, ask the installer or builder, etc. If it was a prefinished floor it probably has a proprietary coating that includes aluminum oxide for hardness. In any case it probably doesn't matter. I'd simply use gloss urethane for this little patch.
What's the direction of the wood fibers (or grain)?
You're correct that it's technically parallel to the length of the board. Even though there appears to be a curve in the grain direction, that's mostly a result of the type of cut made in the board.
How do I stain the wood after sanding down the wood filler and the area immediately around it?
Find a dark walnut or other stain that matches the darkest part of the knothole, and use a small art brush or cotton swab to stain just the patch. Actually, you may find a very dark wood filler that doesn't require staining.
You want to simulate a knot, not try and hide it. Also, you should avoid sanding the existing finish. I'd approach this repair from a hole-only standpoint. Do your best to avoid disturbing the finish on the adjacent area, as it will be very difficult to hide. I'd go so far as to apply masking tape to the area to preserve it.
What grain of sand paper do I need to use for the sanding?
The "grit" you'll use depends on how you apply the patch. Ideally your putty knife will leave a fairly smooth surface. Finish with 120 and 180/220 before sealing.
Is there anything else that I need to know before starting?
Just to reiterate, seek to repair only the hole, and not the surrounding finished surface.
Will the hole deteriorate if I leave it as is?
That's hard to say, not knowing the use of the floor, but it's likely that the edge of the hole will eventually degrade due to foot traffic. However, you shouldn't see serious damage unless it's susceptible to table legs, etc.