Where should be the horizontal braces of the stand placed? In this diy, horizontal braces are placed 17 cm form the bottom of the stand. Is it necessary? Can I place it everywhere or is it needed to calculate where should I place it?
It doesn't matter. My former stand I had under my previous tank was taller and that braces served for placing the shelves. As the legs were shortened, the braces are where it is :-)

May an aquarium be supported by the styrofoam pieces only at its edges?
Yes, it can stay on the stand edges only. I use to use a full support uder the tanks anyway. I cab sleep better then :-) In the fishroom, there it does'n matter but if a few hundreds litres of water will flood in your flat it is not a fun :-) But, the full support board must be absoltutely flat. In other hand, it is much safe to support only the side edges of the tank.

Should be the stand legs supported? It will stay on the lino.
Some steady supports will be fine as they also help to distribute the pressure on the base. A piece of the steel sheet can serve. If the legs are not closed at their bottom side they will penetrate the lino.