As to the rest of your question.
1) Restaurants.  Stay away from the two- and three-star places.  Over priced and snooty.  Stick to the one-stars.
2) Picnic.  Go to the Rue de Buci where there are loads of specialty food stores.  Buy some things you like and walk to the Luxembourg Gardens for a picnic.
3) Walk through the Marais district.  It's the oldest part of the city that's left and has nice shops.

There are literally hundreds of options for the purchase of goods for your picnic in the areas around and between St. Germain and the Sorbonne -- little meat shops, pastisseries, bakeries, flower shops, wine shops, chocolatiers.  In the summer there are large open air markets, but I don't know about the fall.  One of the best open markets is north of Luxembourg Gardens near the Odeon Theatre -- but again, that is in the summer.

The Marais isn't far from the center of the city.  It's only about a ten-minute walk from the Ile de la Cite, so staying there should be fine.  The Marais is also the old Jewish Quarter, so there are still a few delis there and restaurants with middle eastern food.
That said, when I've stayed in Paris it was always on the Left Bank in the St. Germain area where there are lots of what used to be reasonably priced hotels.  My guess is it's hard to find anywhere under $300 a night there.  
Good luck!
A final tip.  Unless you speak really good French, stick to English.  Parisians are generally pretty nice but simply hate it when an American insists on speaking lousy French.

when I tried to pick up gals in bistros by saying:

Come out in the alley, you?

(Comment allez-vous?)

Hint: When you have to explain your joke in parentheses, you probably should not have told it to begin with.
N'est ce pas

This is what comes from sleepy posting as I only got a few hours last night on account of my plane delayed from comimg back from seeing our boy M____

It is cold and dreary at this time of year but still so many good things to see.  Take a day to walk the river.  Start  n the morning near place de Concorde near the seine and Visit the musee l'orangerie and see the water lilies of Monet. Walk along the river to the pont neuf and cross over to the ile de la cite.  Get a bite at taverne Henri IV there .  Stroll the ile de cite past notre dame.  Go in if you want and see the rose windows but also walk around it and look at the flying buttresses supporting the walls.  Then past notre dame to the pont St. Louis connecting the ile de St. Louis.  Stop again at Le flore en l'isle across the bridge and get a hot apple crepe and a glass of calvados.  Maybe some Bertillon ice cream too if you're not too cold.