In the Arena, you can play a chain of 7 matches. As long you win a match, you will keep accumulating points in the chain. The chain stop whenever you lose a match or you win the 7 matches of your chain. Then the chain (and your chain score) will reset back to 0, and you will be able to try to make a better chain score again.
Scores from different chains don't accumulate. Only the highest score you got on a specific chain is conserved, and will be used to determine your weekly arena reward.
The score you get at the end of a match depends on different points:
- The difficulty tier you selected. To each tier is linked a score range. The higher is the tier you select, the higher will be the score range. The counterpart is that the team in front of you will be way more difficult to defeat too.
- The level of your team. The team you will fight are more or less matched with the sum of the stats of all the members of your team, and so is the score range. The higher is this sum, the higher the score range will be for each tier.
- The number of people alive in your team at the end of a match. Your possible score will be maximized if you were able to keep all the members of your team alive at the end of the match.
- If you are using one of the featured characters in the actual arena season, the possible score will be multiplied by 2. Do note that adding more than one featured characters won't add additional points.
For your second question. If by rank, you are meaning your Arena Tier, it is mostly showing your skill in the arena corresponding to what you achieved in the previous season, but it doesn't seem to me that they serve any purpose in the game at the moment. Do note that your rank can only increase by 1 at the end of a season, and only if your score is high enough.