It doesn't matter where they live. Foreign investors can still be required to pay income and capital gains taxes on money earned in a particular country. The government can always block or seize international payments to tax evaders.

In theory the countries where Lewis Hamilton races and earns his income could tax him there. But generally they let him slip through loopholes to maintain good relations with F1.

So Lewis Hamilton needs to fill tax returns in the 40 countries he works in at the end of the season

To be fair: I'd expect Mr. Hamilton to be payed for his 40 races by exactly one company which is based in one location, rather than for each race by one of 40 companies.

This depends on the tax laws of the country. Some countries will tax certain income, other countries won't.