For climate - check weather.com for averages. We were in Paris in April 2006 - it was in the 60s mostly. I layered - usually a long sleeve shirt with a blazer with a scarf/pashmina. Don't think I wore an actual jacket much.
In Paris, you should sit in a cafe and people watch for a while. There are also hop-on/hop-off boats on the Seine. You can get an all day pass and it's a nice way to get from place to place not being on the metro - you can see things you might not have time to actually go visit. We used Batobus. We got our tickets right by the Eiffel Tower.
You must go see the Eiffel Tower. You don't have to go up (we didn't b/c I had a horrible head cold and was so congested I thought my head might explode on the elevator), but you should just go see it, stand under it, grab a crepe and sit in the park there and eat it.
For a wonderful, romantic view, go to the bar in Montparnasse Tower. It has a 360 view of Paris. Get there early enough (before dusk) to get a seat facing the Eiffel Tower. Then when it lights up you get the full glittery show! Beautiful. It costs to go up there, but get some champgane and gaze out at the view.
http://www.tourmontparnasse56.com/uk/
The Louvre takes a whole day, so I wouldn't go to it if I were you with only 2 days in the city. If you really want to go to a museum, we hit the Picasso museum. It was cool b/c it was in his old apartment/home but also small enough that you weren't there the whole day.
You'll have a wonderful time! Enjoy!