Which grammatical case is "him" in "I help him", and why?

English doesn't really have a concept of direct versus indirect object in the same way.

To answer the easy part of the question: both eat and help take direct objects in English.

While there are some verbs (which I'll come to next) that some people analyse as taking both a direct and indirect object, for the most part English uses prepositions where other languages might use a dative or indirect object - so we say things like

I said "hello" to him.

I did the washing for him.

I mentioned your name to her.

Some verbs (such as tell, hand, give, show) can express two objects without using a preposition. It is possible to call one of these objects direct and the other indirect (and indeed people often do - not least because it's useful to have names to distinguish the two objects), but in many ways it is better to analyse them as being ditransitive (that is, taking two direct objects). For example:

I told ¹him ²the news.

I handed ¹her ²the book.

I gave ¹them ²a cheese and pickle sandwich.

I showed ¹you ²the way to use the chainsaw.

That saved ¹me ²a fortune.

These can usually also be expressed with a preposition:

I told ²the news ¹to him.

I handed ²the book ¹to her.

I gave ²a cheese and pickle sandwich ¹to them.

I showed ²the house ¹to you.

That saved ²a fortune ¹for me.

In the case without prepositions, the so-called "indirect" object is generally the one that comes first (though there are dialects in which saying I gave it you is correct for I gave it to you, and similarly for the other verbs).

In the case where these verbs are used with a single object, it is usually understood as the indirect object (the one that would naturally follow the verb even if there were two objects), but this is not grammatical for some verbs, and for other verbs what follows is a direct object (sometimes with a different meaning):

I told him. (single object "indirect")

*I handed her. (ungrammatical)

*I gave them. (ungrammatical)

I showed you. (single object "indirect")

That saved me. (grammatical, but with different meaning)

He threw me the ball. / He threw the ball (single object is "direct")

He brought me a turnip. / He brought a turnip (single object is "direct") / He brought me. (grammatical, but with different meaning)

Why consider both objects as direct objects? One reason is that there is no distinction in object pronouns; "them" is used in both the "direct" and "indirect" positions. English simply does not have a concept of "dative", even in pronouns (which retain more case distinctions than standard nouns). For another: in languages that have true indirect objects, it is not usually possible to transform the sentence into the passive voice with the indirect object as subject, at least not without some sort of rephrasing. This is not true with English, since either object can, in most cases, become the subject of a passive:

He was told the news by me. / The news was told to him by me.

The book was handed to her by me. / She was handed the book by me.

A sandwich was given to them by me. / They were given a sandwich by me.

?I was saved a fortune by that. (questionable) / A fortune was saved for me by that. (grammatical, but with a slightly different meaning)

So, finally, to address your last two questions:

How do you know what kind of object an English verb takes? It may take zero, one or two direct objects, and possibly other complements with prepositions - but you have to learn all of these along with the meaning of the verb; there's no easy way out I'm afraid.

Which case is the object pronoun in the phrase "I help him"? The word him here is the objective case - and indeed him always is; there is no concept of direct or indirect object pronouns.

There is only a slight grammatical distinction between direct and indirect objects in English. Indirect objects can’t be moved around in a sentence quite as freely.

He asked if I bought Kaitlin presents. (ok)

He asked what I bought Kaitlin. (DO preposed - ok)

He asked who I bought presents. (IO preposed - wrong)

I gave her a headache. (ok)

[The headache I gave her] hasn’t gone away. (DO preposed - ok)

[The woman I gave a headache] hasn’t gone away. (IO preposed - iffy)

But there is no such distinction between eat and help:

[The pancakes I ate] were pretty good. (single object preposed - ok)

[The student I helped] was pretty good. (single object preposed - ok)

So grammatically, there is simply no grammatical distinction to be made here. We say that both eat and help take a direct object.

The same is true about verbs like tell that can take a single object with either meaning:

[The secrets I told] weren’t pretty. (single object preposed - ok)

[The man I told] wasn’t pretty. (single object preposed - ok)

Regardless of the semantic role of the single object, it’s syntactically a direct object.

Update — I completely rewrote this after reading the bits of CGEL about indirect objects (chapter 4, §4).