How to best cover a fence post?

Our gate is not that tall, so I can't say whether the way it was done here is an option for you. First, a continuous concrete foundation was made to prevent one pillar from settling differently than the other and causing the gate to become crooked and jammed over time. Where the pillars were going to be placed, elongated recesses were provided in the foundation. The locksmith inserted quite stable U-beams into these. The locksmith and the bricklayer, who subsequently built the pillars with facing bricks, coordinated on the space needed for the bricks. Then the locksmith welded corresponding hinges to these U-beams so that they protruded far enough to allow masonry work and still leave several centimeters of space between the gate and the stone. Of course, the U-beams were also embedded in the foundation. Some time after the pillars were built, they were poured.

Because I haven't set up the fence yet, in the picture it is still at the construction site, but we have taken it down and it is lying around here, so nothing is set up yet, so how should we do it best?

The fence and the gate are purchased, so you like the fence, so put it up. If you don't like it, too bad about the money.

I don't understand what is supposed to be ugly about posts that have the same design as a gate that one likes (?). How is the fence (or the wall?) supposed to look with it - the picture probably shows the situation at the seller (?)

I also think the post is nice. It matches the gate.

How should it continue on the right and left of the gate? Will there be a wall, or is one post at the house and the other in the hedge?

If you want to do it yourself, keep in mind that there are considerable leverage forces at work. If the posts are not really straight, the gate will open or close by itself. A properly cast foundation is mandatory, also in the case with the posts.