I'll try and answer your questions, though there are several approaches to both parts - the joy of Astrology
Firstly on the strongest planet in your chart. It depends alot on what is meant be 'strength'. The Sun is usually seen as the most important of the planets, followed by the Moon. The Ascendant is not a planet but a point, however the ruler of the sign of the Ascendant is also taken as being important - In your case Aries is on your Ascendant so Mars is the Ascendant Ruler - often referred to as the Chart ruler.
The idea of rulership leads to the concept of 'dispositors'. My Sun is in Libra, so its ruler of dispositor is Venus. It's similar to the concept of a guest, The Sun is in Venus' house so it looks to Venus for permission to act. Now my Venus is in Scorpio so I would take its dispositor to be Mars (I work on traditional rulerships). In turn Mars is in Leo, so it is ruled by the Sun. So I have three planets that are interdependent. Indeed most of my planets are in Scorpio, Leo or Libra (with the Moon in Aquarius and Uranus in Gemini). Sun, Venus and Mars together rule the rest and have to get along with each other. So I could come to the conclusion that these three are my strongest planets. For some people there may even by a sole dispositor, one planet which directly or indirectly rules the rest. This could be said to be their strongest planet. In your case Jupiter rules directly or indirectly 7 planets including Mars. However you have Jupiter in Aries and Mars in Sagittarius, so Mars also rules Jupiter - a situation known as 'Mutual Reception'. Mars rules the other planets not ruled by Jupiter. So Jupiter and Mars are the two strongest on this basis, with Jupiter shading it from Mars.
A second approach is to take planets that are Angular as being strong - You have Jupiter on the Ascendant (indeed within a degree of your Ascendant) this would make it strong and some Astrologers would take it, rather than Mars as your Ascendant ruler, given that it is so close). Again it is possible to have more than one Angular planet. Planets on the Ascendant or MC are probably stronger than planets on the Descendant or IC.
Traditionally there is a system of essential dignities that goes beyond sign rulership. It includes exaltation, and rulership by Triplicity, Terms and Face. These essential dignities can be ranked and 'scored' to find out which planet has most essential dignity in your chart. The Astrologer William Lilly also had a scoring system for what is termed 'Accidental Dignity' - that is dignity by where the planet is in your chart - Angularity mentioned above is one form of Accidental Dignity.
Through both Essential and Accidental dignities Lilly worked out what the strengths of each planet were and which was the strongest. This approach is still used in Horary Astrology but is rarely used in modern Natal work.
Ok, so we have three measures of 'strength' - dispositorship, Angularity or Dignity (both Essential and Accidental). These are not the only possiblities but are the most used. We also have 'importance' in Sun, Moon and Ascendant Ruler.
On your second question, striclty speaking they are not imcompatible readings, being successful and honoured does not mean that you are liked
I would tend to take Sun in the tenth to mean that you will express your creativity and your will through career, or the role you choose to play in society - that does not guarantee success or honour - though if sufficient effort goes in, then who knows what might happen. Moon in the twelfth may well suggest that you keep your intuitive side under wraps, it acts behind the scenes. It might also be something that occasionally leads to you getting things wrong - self undoing and yes, it can mean that the public have difficulty getting to know you because you may often need solitude to deal with your emotional cross currents. Your Sun has only one aspect - a sextile to the Moon, so these two planets are more likely to work well together than cause stress (though working well together does not mean the overall effect is nescessarily positive).
Your Moon is disposited by Mars and is trine to Mars, so it will work well with your Ascendant Ruler. However it has a square to Neptune which might suggest a tendency to be overwhelmed by emotions or find it difficult to pin down your feelings. It is also likely to indicate a sensitivity to the suffering of others.