Can I use " had " instead of " if " without having a past participle verb after " had "? In other words, is this sentence correct?
* Had he a car, he'd give me a lift.
That sentence would work in British English, in which the main verb "have" ("He has a car") functions like an auxiliary, just as main verb "be" functions as an auxiliary in British and American English ("He is a driver" --> "Is he a driver?").
In American English, we would say, "Does he have a car?" But speakers of British English can say, "Has he a car?" And this subject-auxiliary inversion can be used in conditional sentences, too. Here is a famous example from the year 1681:
Had we but world enough and time,This coyness, lady, were no crime.- Andrew Marvell, "To His Coy Mistress"
That example is archaic in its use of "were," which nowadays would be "would be" in that position. But the conditional inversion with main verb "have" still works in British English.