Your box should include the camera, battery, battery charger, manual and software.
You will need to buy a Class-4 or Class-6 SD memory card. Though there will be some inbuilt memory, it will be too less to carry on with. Practically any brand can do, but if you want to be choosy, maybe go with SanDisk or Kingston.
A bag is recommended for the camera and related accessories. Choose a size which best fits your needs.
The FZ35 comes with a neck strap, and if you have a bag, a separate shoulder belt shouldn't be needed.
What about this image (the 4th one) has a weird lens. anyone knows what it is? Is it included?
Sorry, but I don't follow you here....
You may want to buy a UV and/or polarizing filters. They are the most essential ones. A UV filter is strongly recommended as it doubles up to protect your lens.
Hope this helps,