5-7 days is more than enough to see the main sights of Paris.
You will be going in winter, so expect shorter days and gray skies. You will be lucky if you get nice blue skies.
We stayed in a hotel near the Gare du Nord station - 2 mins walking distance. We paid Euros 68 per day for a very basic room with double beds and breakfast. Since I was there only for 1 night, can't comment much about the place. But it was recently renovated.
Europe is usually quite expensive. Budget around Euro 30-40 per day on food. If you are going only to Paris (from Singapore), it won't be a bad idea to take a few bowls of cup noodles (if you are on a budget)
The best way to explore Paris is on foot.
Since you have 5-7 days, you can choose a different spot every evening to shoot the sunset - if you are into such kinda things. I am a sucker for sunrises and sunsets.
My advice is as follows :
1 day to see the Louvre (keep this dynamic so that the day it rains, you can hide inside there). Louvre at night is extremely beautiful too.
1 day walk from Louvre through Jardin de Tuileries through Place de la Concorde through Pont Alexandre III through Hotel Invalides through to Eiffel Tower. Do the walk back again at night (you won't be able to walk through Jardin de Tuilieries).
1 morning Sacre Couer + Arc de Triomphe + la Defense + afternoon/evening/night to see Eiffel Tower (the best view of the sunset is from Trocadero). Note the sunset times and stake a prime location at Trocadero (right at the centre of it) 30-45 mins before sunset. Put your tripod there, so nobody can stand there - typical kiasu mentality. Heck, who cares - you don't go to Paris everyday. After that, walk through underneath the Eiffel Tower and on to Champ de Mars
1 morning Pantheon + 1 afternoon/evening to see Notre Dame, St. Eustache, then go to Arc de Triomphe to shoot the sunset and take photos at night.
1 day for Disneyland (if you are into theme parks)
If you have more days, you can venture a bit farther off Paris. I am not familiar with that part, so can't help you much.
I will check my guidebook and let you know if I find anything else.
But the Metro Pass (they have multiple days passes - 1 day, 2 days etc.) Take some time to familiarize yourself with the Metro network. Its complex but extremely easy. Once you get used to it, travelling will be a breeze. Metro runs till 2 AM. Be careful on the RER.
If you need a map of Paris, I can pass you one. If you need any other tips to orient yourself, let me know.
There are lots of books on Paris in our libraries. And you have plenty of time to do your research too. Good luck.