Arena Level is different than Arena Rankings. at certain Arena Levels, you can get a Vanity Aura. there are 7 Aura’s total. after that, Arena Level is just an indicator of how of how much fighting you are doing in the Arena. Ranking is based on your MMR, which changes as you win or lose. the top 1% of players are in Eternal League.
going to higher Map Floors just means killing monsters. Floor 21 and higher have a boss that needs to be defeated to go to the next floor. the highest floor is 200. Challenge Maps have a Cartographer (the boss) that needs to be defeated to get a map for a higher floor. you need to defeat Cartographers in Challenge Maps to go to floors 201 or higher. the higher the floor and the harder the difficulty, the better you need to improve the damage you do, do the damage faster, and stay alive while you are doing the damage.
it looks like you are still in the early stages of playing DQ, so one bit of advice is if you are farming for loot to make better items, Farm on a lower difficulty level. the faster a monster is killed, the better/more loot it drops. as you get better loot to make better items, you can improve the difficulty level to get Legend items that can only be gotten on the higher difficulties. also, if you are going for the Perks to improve your farming ability, you only need to kill monsters on floors 100-110. experience from monsters stops going up at floor 100, so if you go to floors that are too high, it takes longer to get experience, even if you are just leveling up your Character.
take a look around the Forums for posts for Beginners and New Players.