You should match Caput Algol onto whatever houses Jupiter is ruling (houses with Sagittarius and Pisces on the cusp) and apply it to them. The person should be aware that the issues of the star of distraction, debilitation etc. Will all stem from those houses.
Jupiter itself is an odd one for fixed stars, it likely just increases the energy of the fixed star in general. If you did any planetary square though, I believe it works on your current transiting astrology/progressions- not your natal chart. You should be fine with Jupiter Squares to help alleviate problems from it. The natal chart is the core blueprint, motion charts are the moving blueprint, wherever Jupiter is for the moving blueprint is what acts on your soul so yeah I wouldn't do a square if it was currently over Caput Algol. Natal chart presence should be fine to do planet workings though.
For your other question, it's a case of the chart ruler and Mars. Mars is the natural ruler of the 1st house, so issues in the first house can sometimes be fixed with Mars workings. If the problem is lack of self assertion, identity, or laziness, then being a natural first house issue- it will benefit from a working on the natural first house ruler. If it is something else pertaining to it, you would have to work on the chart ruler.
I said in some other post as well, for outer planet rulers, they are connected with parts of some chakras. Uranus/aquarius "co-rules" the root chakra by default as well as the aura. Neptune rules the right side of the pineal gland and co-rules the crown chakra. Pluto rules the left side of the pineal gland and is connected in a big way to the sacral chakra. For personal planet rulers it's easier you just get the chakra associated with the personal planet and boom, sorted.
The squares strengthening the natal placement is to do with the core blueprint analogy. If you swap copper in the support and replace it with titanium, it's going to be stronger. It will be more resilient to other forces, which is strengthening the soul against outer forces. Fixed stars are useless against Titanium, and when an SS has maxed out a planet which was burdened in their natal chart, they have control over it. It doesn't strengthen the people, cars and tanks(Caput Algol) crossing over the bridge when the bridge turns into Titanium. The SS' planet and the force of the star acting on it are separate. Unless theres all of a sudden more tanks or they start firing and blowing up the bridge (exploding stars, lol)
When the bridge is getting blown up by tanks, you probably shouldn't be turning it into Titanium then, you shouldn't do workings when planets are in a bad placement currently.
I actually don't think titanium is good for building bridges, lol. Way too heavy.
