I'm pretty new at modding TS4, and I tinkered around with some XML files this weekend. I was somewhat able to get dogs to go on top of beds by giving them access to cat surface areas.
1. When big dogs jump on the bed, their hind legs will "land" beyond the edge of the bed, making them appear as if they're floating on air for a short while.
2. I'm not sure if dogs would climb beds autonomously since this was achieved using the controllable pets mod.
3. Dogs can also jump on some other surfaces, like cat condos and the washing machine, which might seem odd depending on your preferences.
4. Dogs can't be instructed to sleep on the bed (or even when on the floor for that matter). I simply get an exception that the MCCC logs. So at best, dogs can sit and lie down on human beds.