Can a sink drain be on the low side of a p-trap
Replacing the whole lot is certainly the way to go. The layout is daft and it seems held together with wishful thinking.

What to do?
Something like what you have in mind but IMO the dishwasher should join in immediately above the p-trap.

Many companies sell complete packages for what you're trying to do. 

E.g.. 


(source: plumbers-mate-sales.co.uk)  

Or  this one listed as "space saving"


(source: scene7.com)  

Unfortunately, once you've done all of that I think you're still going to to find that the waste pipe comes out if the wall a little to high up.