Can the type of handrail shown in the photo be mounted onto a wall using brackets for use in a stairwell? As you can see it has a groove cut out of the bottom.
Maybe, but it is not a good choice. Like Kris' comment said, this is made for the top of spindles.
It will look funny IMO.
You may need to notch the inside to accommodate the wall bracket (depends on the exact wall bracket that you use.
Since the vertical dimension is fairly thin, it will be less rigid than expected. You could compensate by using more wall brackets than normal.
If I really wanted to use this profile, I would put a steel bar in the groove running the entire length if possible, or if necessary, use several with the breaks near the brackets.
There may be building codes that I am not aware of. Common sense says that a strong man should not be able to pull it off the wall.
Yes, that is a Type 1 handrail and is legal if: 1) it’s at least 1 1/4” in diameter and not more than 2”, 2) if it’s not circular, (which yours is not) it must have a perimeter of at least 4” and not more than 6 1/4” and a cross section dimension of 2 1/4”. (See ICC R322.214.171.124)
Type 2 handrails are larger (like made from 2x6 trim) but must have finger grips. (See ICC R3126.96.36.199.2)
Make sure there is 1 1/2” clear between the handrail and wall.
Make sure the handrail “returns” to the wall or extends to the ground or post. That means you need to miter the handrail at a 90 degree angle so it fits tight to the wall.
Handrails are not required on stairs with 3 risers or less.
Mount the top of handrail between 34” and 38” above nosing of tread.
The hand rail you chose will work with practically any handrail wall brackets on the market, if you choose to use that type. Do a search online for "handrail wall brackets" You will see the part that attaches to the handrail is basically the same on every style bracket.
The plow or plough is for a fillet piece that is cut between the square top balusters and should come with the handrail, at least any plowed rail I bought, it came with it. I have had to add the fillet to a few rails to get past some details on jobs.
You should have no problem finding the same rail that is not plowed, try that first, it will look better.