Can I texture and paint over existing wallpaper? Or do I need to peel the wallpaper off before texturing?

If the wallpaper isn't peeling and in good shape you can after filling all holes and priming with an oil-based primer.
Read this link:
Texture How-to Guide (http://www.wel-cote.com/PSSHTM/Texturebook.htm)
I painted over the wallpaper in my bedroom after priming and it still looks good after 7 years.
Use a satin or semi-gloss paint to seal the texture.

Patty is correct on the most parts of it you do not have to prime with an oil base paint you can apply drywall mud right over the top of it if the paper bubbles a little take some of the drywall mud and put it behind the wallpaper and press it back the mud will hold it there texture and paint what happen when you paint it first the drywall mud bubbles and you have to go over it 3 or 4 times

The best method to use is to seal the wallpaper with a cover coat of a product like Zinnzer Bin Sealer. Dries quick enough as to not lift or bubble the wallpaper. 1st coat light. 2nd (if needed) apply normally per can instructions. Also covers up stains. There are other "BLOCK-OUT" sealers at your local home improvement centers as well. Most new ones are based with safe ingredients. Ask the paint people there. Zinnzer is paint ready to paint in 45 minutes. Strong odors possible so use adequate ventilation. I have owned/rehabbed 5 houses. Have run into pretty much everything. Lightly patch over seams/tears if you don't want them to show.