(1) Yes. The dummy es (Korrelat) can frequently be omitted. However, as you seem to have observed, German speakers often have (sometimes strong, sometimes weak) preferences about the Korrelat use.
The preference is generally related to the information packaging in a given sentence (Duden-Grammatik (9th edn), para 1706), which in turn is partly verb-driven and partly context-driven. A factor that causes speakers to disfavour the use of a Korrelat is if by the time we reach the (hypothetical) midfield, there is still a huge informational gap. Your example Ich lerne is vacuous in and of itself, it conveys barely anything, and tells you nothing of value. A more extreme example might be something like Ich versuche, as in Ich versuche, eine gute Note zu erhalten. In that case, I believe that most speakers would probably go so far as to consider the use of an es Korrelat agrammatical. Metaphorically speaking, you don't want to close the lid on this sentence prematurely, not even by just a few centimetres, until the infinitive construction provides the much-needed information. Examples of verbs that hardly ever come with a Korrelat are befehlen and beschließen.
An interesting - context-specific - illustration of this tendency is given in the following example (from IDS-Grammatik II, p 1484): Es ist schrecklich, wenn vor so vielen Dingen ein dunkler Vorhang ist. Ich möchte ihn immerzu zerreißen, aber ich kann es nicht. Ich glaube es dir, daß du diesen Vorhang nicht zerreißen kannst. Note that glauben is one of the verbs that are rarely seen with an es Korrelat, so it is worth noting what might set this sentence apart. As Zifonun et al observe, "the Korrelat here has, in addition to its forward-referring function, a backward-referring function with respect to something previously addressed or established in the [speaker's] knowledge".
Unlike lernen, I would argue that in terms of informational content Ich hasse is a statement that already conveys a substantial amount of information. The point of the sentence is to convey an emotion and Ich hasse does that job quite well on its own. Saying that you love or hate is not devoid of meaning, unlike beginning to describe an activity (lernen) without actually saying what that activity is. Anyway, the usage differences are not entirely understood as yet, so you are touching upon an issue that is difficult to briefly explain in a forum like this.
(2) I'm not sure I understand the question but I think I might have addressed it in my answer to (1). In a sentence like Ich lerne, den Ball weiter zu werfen, there is a strong preference not to use a Korrelat es. But would using it be perceived as agrammatical? I certainly don't think so.
(3) Yes, see my answer to (1).