Nothing. The starting dungeon in which you find yourself is
designed to help you advance in levels before venturing out into
the real
world. USE IT! There are numerous caches of money,
weapons, and magical items. If you are careful exploring this area,
you can leave the dungeon probably around 3rd level. One last hint,
areas which are raised off the floor are safe to rest in, provided
there are no monsters currently alive and within range of you
Quests are earned based upon your reputation and level. At low levels, Kings and Wueens will seldom look to you as their champion. Be patient and work hard. Once you begin getting into higher ranks, your reputation will proceed you and you will find that rulers of different cities will try and enlist your aid.
Take the number required for your class to reach 2nd level, then multiply that by 1.875. This is roughly what you need for the next level after that. The reason that we say roughly is because the actual multiplier varies a bit from subclass to subclass, depending upon their special abilities.
Remember that the spell effect has some space associated with it. Although it probably isn't in your best interests, you need to give the spell some room to fire. Try (carefully) sliding out a little further before casting.
Yes. While it is certainly true that certain classes have advantages over others, they are balanced out by needing more experience points at each level. As a result a Thief class character will generally be of a higher level than say a Spellsword. This directly translates into a better chance to hit an opponent, better saving throws, and higher base stats. It is imperative however that you adequately prepare for a dungeon by stocking up on potions and magic items. They can make the difference between life and death. We know there are a number of ways through each puzzle / area in the game that are achievable by any class. in your travels you will probably come across others.
If you want to try your hand in a dungeon quickly, (besides the one in which you start in) there is one located just south of Ebonheart, in the Morrowind province. Good luck