Yamaha R1 2008 Coolant pushing up through radiator cap when bike warming up. Also noticed coolant on left side of engine coming through casing

Typically they join in near to where the cap sits on the radiator.

When the cap opens due to either high or low pressure in the radiator, the coolant will flow either into or out of the overflow bottle, so the plastic lines need to be close to the cap.

Most likely a blown head gasket. Exhaust gas is blowing past the gasket and into the cooling system. It will pressurise the cooling system and push the coolant into overflow bottle. You might notice that after riding the bike for a while and then leaving it to stand for a few minutes, it will push coolant back into the cylinder and you will get a big puff of steam out exhaust when you start it.
Best way to check is to do a compression test. Need to remove spark plugs and and screw a compression tester into the hole. Crank the starter and you will see the gauge go up in 3 or 4 strokes. Should be about 224 psi for a new stock motor. Repeat test for remaining 3 cylinders. Pressure should be 90% of that figure (201) and should not be more than 10% different from highest to lowest. Probably find one cylinder is really low.

Hi,,,,check your coolant level,,,,also might have an airlock,,,take off the radiator cap when the bike is cold and start it up let it warm up and make sure there is coolant to the top of the radiator.

check that the radiators are clean from mud ect,that blocks airflow.

There should be a coolant overflow bottle behind the side cover on the left side of the bike, near the seat. This needs to be filled to between the low and high marks. You also need to ensure that the radiator itself is full. The cap is on the top of the radiator on the right hand side of the bike, underneath a plastic cover from memory. Fill this right to the top.

All this needs to be done when the bike is cold to get the correct settings.

Nice bike by the way, there aren't many other bikes that perform so well regardless of what kind of riding you do on it.

50 % water 50% coolant

Not sureon your year of bike, but start with 1 quart, then wait a few minutes, check, and add from there.

Howdy: If the air temp is hot and raining the engine will heat up somwhat but not to the hot level. There is a temp sensor in the radiator system (refer to your manual) that may be stuck. Do a flush and pour in new coolant. Also when engine is cold undo the radiator cap and see if the coolant starts to move as the engine warms up. If not water pump rotor may be loose or broken and not pushing the coolant thru the engine. Try it - Good luck and ride safe - Doug

Air is gettiing trapped in the radiator/engine block.

The header pipe will glow red when you run it on choke, during warm up. This is normal while sitting. You shouldn't see that one you start riding it though.
The coolant drip could be just a dirty or failing radiator cap. My bike will push a little out if I completely top it off full. Hope this helps.