Gfrktyl

This is what I've learned so far.

Ob/wenn::
both means "if" but use "ob" to convey a sense of "whether".
If not, you must always use "wenn". Only one is right in any given sentence.

Falls:
means if, can be used instead of "wenn" to express possibility, it is more tentative(unconfirmed) than "wenn"

So for this sentence or sentences similar to this one, which one is grammartically more precise:

Did you want to play football? If yes, with whom?

1. Wolltest du Fußball spielen? Wenn Ja, mit wem?


2. Wolltest du Fußball spielen? Falls Ja, mit wem?


3. Wolltest du Fußball spielen? Ob Ja, mit wem?

I think the third one is wrong. I would be grateful for the explanation with examples and grammar behind it. Thanks a lot in advance.

Wolltest du Fußball spielen? Wenn Ja, mit wem?

Wolltest du Fußball spielen? Falls Ja, mit wem?

Those are both correct, the former being the somewhat less formal and the latter being the somewhat more formal form (in the South you may be perceived a bit snotty when saying "falls", in the North I'd say you are OK).

Note that although strictly that isn't the case (they're equivalent!) many people will imply an at least indifferent or positive-ish attitude with "when" whereas with "falls" the baseline assumption is rather negative.

That's probably because that implication is truly existent in the other use case of these two words:

Wenn wir ins Kino gehen, will ich Popcorn.

Falls morgen Fußball gespielt wird, komme ich nicht zum Spiel.

Here, the first sentence (weakly) implies that we're going to cinema (this is not much doubted), and when that happens, I'll be wanting popcorn (when, not if). The second sentence implies that it isn't certain at all whether or not we will play soccer tomorrow (but if we do, I won't come anyway).

Wolltest du Fußball spielen? Ob Ja, mit wem?

This one is, uh... strictly "correct" but very, very antiquated and awkward. Unless you are reciting a Psalm from the Bible or are a medieval author, you will be given a puzzled look if you say such a thing.

Interestingly:

Wolltest Du Fußball spielen? Und ob!

Here you get the meaning of "well yes, of course, most definitively", and it isn't awkward at all. It may also express strong disagreement with a negative statement or question, such as "I didn't think you wanted to play...".

All your definitions are correct. Just apply them to the sentence and you will see, that you are also correct in your assumption that

Ob ja, mit wem?

is wrong. And your rule, that ob has the meaning of whether is telling you this, since

Did you want to play soccer? Whether yes, with whom?

is wrong in english as well.

Ob ⇆ whether/if:


Only used for simple Yes-No conditions, if the condition were directly rephrased as a question, you could answer Yes or No.
If you can add "or not" at the end of your sentence, it's a job for "ob":

I don’t know, if/whether I have time (or not).


Ich weiß nicht, ob ich Zeit habe.

Thomas is not quite sure if/whether he should drink any more beer (or not).


Thomas ist sich echt nicht sicher, ob er wirklich noch Bier trinken sollte.

Wenn ⇆ whenever/when/if:


Can be used for all types of conditions, but Yes/No Conditions. The goto word for "if".

Falls ⇆ in case/if:


Used when you want to completely rule out meaning "whenever/when" at all. Almost always used only for 'open' conditions, i.e., where in English you do not use "would" because either the condition is realistic or implies it happens.

If it’s raining, I will stay at home.


Falls/wenn es regnet, bleibe ich zuhause.