Waste pipe from kitchen sink to drain question

Sorry water will not run uphill, you need to rethink it.
Hard to say exactly but one of these traps, http://www.screwfix.com/p/mcalpine-appliance-trap-white-40mm/81714 with a new plughole fitting like this, https://www.victorianplumbing.co.uk...n=googlebase&gclid=CP7bsrOSgdQCFQW3GwodqXcOVA might get you out of trouble.

 

With the above you are aiming to get a sink drain fitting the trap fits directly without any adaptors.
By the way the existing waste is solvent weld, compression fittings like you have will fit but pushfit ones will not.

 

Yes, if you try to drop down and have no fall to the stack it will not run well and you could have"things" from the soil stack running back into your sink waste

 

As a last resort you can pump it up & away.

 

A shallow bath trap would tend to suffer from the trap being emptied by syphonage/suction when the WC is flushed. they are normally only used where the bath waste is separate as in the case of discharging into a hopper head.
You could try an elbow on the bit coming off the sink to a running trap like this http://www.screwfix.com/p/mcalpine-running-p-trap-white-40mm/47605 then insert one of these in the pipe between the sink outlet and the trap.
http://www.builderdepot.co.uk/plumb-out-kit.html?gclid=CIWipNK0g9QCFUO3GwodDxUEVA I am assuming the horizontal bit of pipe you were thinking of cutting is standard size.
Ain't plumbing fun!

 

May actually be easier to start afresh, run with all new pipe and fittings, core through wall and run new pipe out to drain

Post a picture of the outside set up and where pipe currently drains to

No point lashing up as will cause future problems, what's the point eh ? Better a bit more money now, materials, effort, etc then job is good for next few decades

 

Being diy I doubt the OP will have the ability to drill a hole that size through the wall for a start never mind make the connection to the drains. If it matched up originally then it can be done again.