Haven't done missions in an absolute age, but I'll try and help.
I'm pretty sure you must be eligible to do lvl 2 missions with standing of 7.
Standing (to progess on lvl missions) is with the corporation of the agent, and not the agent.
In the station of the agent, in the top right should be the logo of the corp from that agent. Click the "i" info icon and you should see the corp's details. Then select the agents tab, and open up all the windows containing different type agents (surveillance/internal security etc.) It will show agents available to you, so you can go and find one to do.
In your present state, you might be able to use some lower quality lvl 2 agents as opposed to perhaps this higher quality lvl 2 agent in the same station. (Are you with spacelane patrol btw?)
The missions you receive by evemail are probably storyline missions which have a greater standing bonus to the faction/corporation, so worth doing if you want to progress. They appear after completing 9 or 10 normal missions.
Caracal should be fine for level 2 missions, just make sure you check your fittings with people in your corp (or here), so you make the most of your ship's potential (dont armor tank it or fit lasers to it

)
Hawk/Harpy tank better with resistance bonuses, move quicker, track better etc. They are great for popping frigates/cruisers, but obviously take a lot longer to train for than a cruiser. They would be absolutely fine in any lvl 1 or lvl 2 mission and could probably do some of the lvl 3s (some have web towers which will muck you up as you rely on some speed to avoid damage).