1.5 - 2hp good than enough your free space 

cooking area far away from living hall, should not be affect the cooling.

I suggest separate cooling for dining and kitchen. High ceiling good to use suspended cassette unless you want to preserve characteristics of ceiling ie left untouched cos if plaster it would be lowered to lowest acceptable height. Hence the suggestion to separate otherwise it would be a waste not to use high ceiling just because plasterinzing it low to cover AC ducting and drain pipe. Alternatively use a wall split model for dining which allows retaining high ceiling feature.

Kitchen AC would be good to use Sharp with Plasmacluster. That helps traps oil and cooking fumes. Fotile has a hood which has this feature also. And an external filter like how kimsim uses it from Daiso prevents the filter from getting dirty so fast. Another AC would be Hitachi with UV fresh and stainless steel filter.

My company provided us an air cond in kitchen. We always turn it on while cooking
Nothing much, every month we are complaining not cold and no air flow  
3 China men that like to cook with big fire, so with a good exhaust also the same = useless
Do not turn on air cond when cooking, as it will leave oil stain and very hard to clean.

Mine is open concept kitchen as well, we do not use it while cooking. Using AC while cooking will strain the AC compressor, as you're heating the ambient air inside your house while making your AC work harder to cool it down. Just make sure to have good ventilation and also a good kitchen hood or extractor fan while cooking.

Yes split vs window type. The former has blower and compressor split whereas the latter is interlinked. I thought so too that you might still like the high ceiling. The volume above the ceiling needs not be cooled. What's more important is the air surrounding the room up to 8-9ft. That's where most occupants are and it's key that they feel the cooling.

If one AC place it on one side of the wall where it faces the kitchen. Here's where long throw blowers like Mitsubishi (Electric and Heavy Industries) and Samsung really comes to play. The long throw mode can reach the kitchen so long as the circulation path is not obstructed. Normally, installer would place it at 8-9ft high.

If two dedicated AC normally because room orientation and placement limitation, Sharp gull wing or Samsung triangle can consider. But Samsung needs 4 core wires vs the rest that needs 3.

Got, the spread of air circulation emphasis is different. I don't know which one is widest or narrowest side but consider this:
On narrowest wall facing widest side , LR circulation not much as UD.
On widest wall facing narrow, LR emphasis depends on cooling zone. Long rectangular table may need more LR or fixed focus L and R. Circular table not as bad.

Then again, the other factor that will influence placement is the length of the run for embedded piping and drainage which implies the amount of hacking needed.