Check power connections and relays ---What is issue / not turning over --slow turning - not starting etc generic answer load test starter may be bad new doesn't mean free from defects - connections are clean and no damage bolted correctly
See if it is coming from the rocker cover above it. If not, it is oil off the cylinder bore that is seeping out instead of being burned. Replace the exhaust gasket.
https://kawasaki.ph/motorcycles/barako-ii/barako-ii/
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Ideal spark plug gap varies slightly depending on brand - check the online application catalogue for the brand of plug fitted to your bike and the recommended gap will be found in the relevant entry.
Yes, there are specific sockets for spark plugs, usually a deep socket with a rubber insert to stop the plug dropping out. For the ZX bikes you'll need a 'thin walled' type. eBay does them as a set for a sensible price.