Let's 1st understand what these planets represent on their own?

Mercury - Mercury represents communication skills, marketing, calculative ability, younger siblings, hobbies, skills with hands, intelligence, quick decision making, logical thinking, youthfulness etc.

House Ruler - Besides these karaka things, Sun & Mercury will also rule some houses in your chart and will represent things related to that house. Of course we can't cover conjunctions Ascendant and house wise else it will never be over. So, you have to include those things at the time of interpreting this conjunction. Also, we need to keep in mind all Yogas and Raj-Yogas which their conjunction would be making to completely understand the conjunction. Here, we can only cover meaning as per karakas. 

Meaning of Sun-Mercury Conjunction - This is normally known as Buddha-Aditya Yoga in Astrology as Mercury is Buddha and Sun is Aaditya. But this is a very common yoga because Mercury & Sun can be about maximum 23 degrees apart. So, 90% times they are in same house/sign. So, this yoga is normally seen in charts. They say that person with this conjunction will be highly intellectual as intellect of Mercury is further enhanced by brightness of Sun. But we don't see 90% intellectuals around us. Then? It depends on how close is conjunction degree-wise. Anything within 10 degrees is good conjunction to show an intelligent person. It doesn't mean that all of them are Einsteins but it only means that person will grasp things faster than others. They can be always looking for information. Mercury is one planet which is not really impacted by combustion by Sun as it is always near to Sun and used to heat of the Sun, unless it is at exact same degree. In that case, person may have issues in communicating his ideas and in matters of relations with siblings. This conjunction shows who can have a very firm and fiery way of communications as Mercury is receiving fire of Sun. Even their writings can be fiery at times. In job environment, they won't be liked much because this way of speaking. As Sun is Career and Mercury is Communications, their career can be related with Communications and Counselling. Even Father can be in similar communication based Career. Or they may have to travel a lot due to Career. Now, something funny. Sun is Brain and Mercury is Speech. If Sun is before Mercury degree-wise, it shows that first brain started processing the thoughts and then person spoke. But if Mercury is before Sun degree-wise, then it shows person spoke and then thought about what the hell I spoke? lol. This situation can be more evident when Mercury is a house before Sun. This conjunction can also give ego issue with siblings and cousins. 

Who will win the Conjunction? - Of course, Sun will. As Mercury is a neutral planet, it changes itself as per the planet it is sitting with. So, with Sun, it is becoming like Sun. Even in Mercury MD, person will get results of Mercury along with Sun's results. 

Dignity & Strength - 

This conjunction will work best in Gemini, Leo and Virgo as both planets are in good dignity here. In Gemini and Virgo, most common Career is Accountancy/Finance related. 

Aries & Scorpio - Sun will wield its authority over Mercury as Mercury is in enemy sign. 

Taurus & Libra - Conjunction may lose its impact as Sun is in enemy and debilitated sign respectively. 

Cancer, Sagittarius and Pisces are another signs where this conjunction can be good as dominant partner Sun is in good dignity here. Mercury debilitation in Pisces never hurt anyone significantly. 

Capricorn & Aquarius - As these signs are ruled by Saturn, the good results of this conjunction will be seen in 30s.