Paris districts

The lower the number, the more central the arrondissement. Google Paris districts or arrondissements and you will get a ton of info on all of them, including the 2nd.

What's as important as which arrondissement is access to "good" metro lines - the ones that will take you to the places you expect to go to. The closer to the metro station, the better. After a day's sightseeing, or a long evening walk, you don't want to have another 15 minutes or more from the metro to your bed. If there are at least 2 lines, going in different directions, or if you are close to 2 metro stations that's usually good.

I haven't stayed in the 2nd but no reason not to. Certainly central, and I'd think lively enough, if you mean cafes, restaurants, bars, street scene, people-watching. Use "street view" on Google Earth to check out the sidewalks. Agree with others about Metro stations, there are plenty in the 2nd.

You don't want to be so close to the Metro that the trains shake the building, however.

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It's not entirely true that the lower the number the more central the arrondissement. Some parts of the 15th arrondissement are very close to Eiffel Tower, for instance.

That may be true Sean, but the Eiffel Tower isn't exactly in the center of everything.

Try www.clickappart.com I have exchanged info with them and almost rented but plans changed.
They do have some reasonable studios for your time frame.