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

House Ruler - Besides these karaka things, Mars & 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 Mars-Mercury Conjunction - Mars & Mercury are natural enemies to each other. Reason is the difference in their basic nature. Mercury wants to be everywhere, wants to collect all the information in the world, it accepts everything. On the other hand, Mars wants to finish the task given. It likes action and precise strike on target rather than having too much information on hand which may confuse Mars in taking action. So, when these two come together, it can lead to a confusing situation where person's actions may be scattered everywhere as he has too much of information and ideas. It means that their actions are not well directed. This situation will be there till Mars attains maturity at 31. Then person realizes the value of focused and one-dimensional actions. Also, Mars is Anger and Mercury is Communication. So, when these people get angry, they have no idea what they speak out in that moment. Mars represents Technology and Machinery. Mercury represents Language. So, they can be very good at understanding languages related to new technology. When they communicate, it is with a warrior-like approach. Even when they write, they can display all their anger and frustration through their written words. They may not have too good relation with their siblings. Overall, if they can keep control over their anger and be focused about their actions, then it can be a good conjunction. But as I said, this situation comes in 30s. Till then, they can be anger-prone. In 30s, they can be very precise about their communications. Similarly, these people should be careful while travelling as Mars association with enemy Mercury can cause accidents and injuries. 

Who will win the Conjunction? - Mars will. Mercury is a neutral planet. It changes itself as per the planet it sits with. So, with Mars, Mercury's communications will be anger-oriented. 

Dignity & Strength - 

This conjunction can work best in Capricorn as Mars will be in exaltation and Mercury will be in friendly sign. Person will be able to control aggression of Mars.

Leo is another sign which can work well for both as both will be in friendly sign. 

In signs like Aries, Scorpio, Sagittarius and Pisces, Mars will be in better position than Mercury. Hence, person's communication or intellect can be misdirected. 

In signs like Taurus, Gemini, Virgo, Libra and Aquarius, Mercury will be in better position than Mars. Hence, person's Anger, Aggression or Actions can be misdirected. 

In Cancer, both are in bad position. So, person needs to be very careful about his actions, anger and communications.