If the first sentence is possible, I guess, the second one can be re-write as “I’d rather you not go out alone." Is this really possible?
Yes, that rewrite is perfectly acceptable. It is however ‘rather’ (in the ‘somewhat’ sense) an old-fashioned way of expressing it.
LONGER (BUT STILL NOT NEARLY LONG ENOUGH) ANSWER:
Expressions with rather are quite complicated, because they are really ‘fossils’ from earlier states of the language. Some of the issues involved are:
The had in your first example is the full verb, not an auxiliary. Grammatically it is a rare instance of the past form representing contingency or unreality outside a conditional construction. Today we express this grammatically almost exclusively as would have; but well into the 18th century the form (which it is hard not to call ‘subjunctive’) sometimes represented the sense ‘to desire or wish [someone to do something]’ (I had rather that ye never maryed in yowyr lyffe - 1478) and at other times the sense ‘to experience or suffer’ (He had better been drowned - 1665).
The ’d in your second example would be understood by most speakers as a contraction not of had but of would, which since the middle of the 18th century has been far more usual than had with rather:
That would, again, represents what is now a fairly rare use: although its past form does as usual express the ‘subjunctive’ sense of contingency or unreality, the base sense is volitive, wish or want. (Would that it were so!)
The verb forms in the complement are also archaic in the cases which employ what you call the ‘root-verb’. I suspect you are being very diplomatic and carefully avoiding calling it either an ‘infinitive’ or a ‘subjunctive’; if you are not, you should be, because sometimes it is definitely an infinitive and at other times it may be called either infinitive or subjunctive.
Without a pronoun, expressing what the subject would rather do, it is definitely an unmarked infinitive, the final verb in the verbal chain of the main clause, not a component of a subordinate clause; this is possible only with the would version, which is grammatically an ordinary modal:
He would rather be dead. derived from He would be dead [=He wants to be dead] derived from He will be dead.
But with a pronoun, expressing either what the subject would rather do himself or what he would rather someone else do, the verb is a component of a subordinate that-clause. Grammarians differ over whether this should be called an infinitive or a mandative subjunctive:
I had/would rather [that] you be dead than that you marry her.
In any case, the version with the past form—“I would rather you were dead”—has been preferred at least since the early 19th century:
- That version of course, must be inverted in a negative, and needs DO-support if the tensed verb is not BE or an auxiliary:
I would rather you did not marry her.
So the more frequent use would be that with the past form and *DO-support: “I'd rather you didn't go out alone. “