Can you install a bidet like a regular toilet?

No, I can't install a bidet like a regular toilet.

both are needed in the same room ... do you have the space?

its normal for a bidet to be placed next to toilet ..every bidet i have installed or seen have built in trap ..just connect the 32mm pipe to soil stack ..and hot and cold feeds ..you dont sit on a bidet ..you straddle it 

Not exactly. They are more like a low level washbasin. It needs a hot and cold water supply and non-return pipework, and a smaller diameter drain with its own deep seal trap. But they are easy enough to install and the drain can link to the wc branch.

Think of it this way.  A bidet is a lavatory for your bottom, not a commode.

no, a toilet will have a 3 or 4 inch drain pip[e but a bidet uses an 1 1/4 inch drain like a bathroom sink does

No .  A bidet does not have a built in trap