Can a private pilot refuel his small private airplane internationally without passing through immigration or customs?
While it depends on the country, expect to have mandatory immigration checks on first port of entry in most countries. I know this to be true for the US, Canada, Britain, Russia and the EU (Schengen area). Usually only emergency landings are exempt. But they may lead to investigations so not really an alternative to declare one in each and every country.

Usually it's best to hire a service provider to prepare all the paperwork for your trip. While their main business is to prepare commercial flights (non airliner), most also accept private pilots as clients, a few even welcome them. They will not just plan your journey regarding visas, landing and crossing permits (some countries do require preregistration for each and every flight into their territory), but also make sure to pick only airports with the right fuel available, or making it available. Selecting and booking hotels can be included as well.

Anyone I know, who ever tried to fly further than the next door country, sooner or later took such service as it makes a huge difference. Well, some had to make the experiance to get stuck for a week at some border first :))