Tense agreement in conditional statements: "I could do whatever I want" vs. "I could do whatever I wanted"

Any of these may be correct; which is correct will depend on what you mean.

Had in the conditional sentences and could in all the sentences are inherently ambiguous. They are ‘past’ in form, but the form is employed in the first instance to express some degree of uncertainty or contingency. Consequently it cannot be known without further context whether they are also being used to express tense—temporal location in the past. Let’s supply a context, to see the difference:

I wish I had my own place. I want to throw a party next week. If I had my own place, I could do whatever I want—I could throw a party. I think I am going to get my own place and throw a party next week.
OR
... If I had my own place I could do whatever I wanted ...

In this case, where all the action is located in the present, either is acceptable in any register. You would be more likely to employ the non-past form in formal discourse, to maintain strictly logical time reference; but in colloquial use you would be more likely use the past form, because all those past forms press for conformity.

In the following case, however, you must use the past form, because the “wanting” is located, with all the other action, in the past:

I wished I had my own place. I wanted to throw a party the next week. If I had my own place, I could whatever I wanted—I could throw a party. So I got my own place and I threw a party the next week.

The same is true of your second example:

I want to throw a party next week, so I talked to my landlady today. She said I could do whatever I want. So I’m going to throw a party.
OR
... She said I could do whatever I wanted ...
BUT
I wanted to throw a party the next week, so I talked to my landlady. She said I could do whatever I wanted. So I threw the party.

It might be noted, however, that in the non-past example you would be even more likely to say “She said I can do whatever I want”, because that permission is non-past as well, and there's no contingency that interferes. You might even say “She says I can do whatever I want”.

All four are correct. 1a and 1b mean the same thing. And 2a and 2b mean the same thing.

This question suggests exposure to a grammar book or teacher who lays out all the rules and says they're inflexible. Alas, that is not true.

First, the "rules" in those books are simply wrong. You shouldn't trust them. They promote ignorance and mythology, like the idea that English has one single construction that is called a "conditional", instead of dozens and dozens of modal constructions, expressing various different degrees of (im)possibility, (im)probability, (un)likelihood, and (un)desirability of some eventuality, and its effects.

Second, grammatical rules are not like traffic rules in a big city; they're more like a map of the city. They lay out routes you can use, with lots of possibilities to choose from at every turn. There are also a lot of one-way streets, dead-end alleys, strange twists and turns, loops, cul-de-sacs, tunnels, subways, trains, buses, taxis, and footpaths one can take.

In this particular case, apparently there is a zombie "sequence of tenses" rule in the woodpile. This is a good example of something that isn't a grammatical rule at all, but just two different routes to the same place.

In the "a" sentences the present tense of want is used in the last clause (the one starting with whatever), and in the "b" sentences, it's the past tense. The question is which is "correct", and that seems to assume that at least one of them might not be "correct".

There are two minimally different interpretations of the "a" and "b" sentences, both of which contain the idiomatic clause whatever X want. Both of them get to the same place in terms of meaning, by pretty much the same route.

Native speakers say both of them, and usually don't notice which one, or even that there's a difference. But when they're written, questions arise, as they always do, because writing is inflexible while speech is extremely flexible.

My favorite example of where writing gets in the way of language is how to write the negative of used to /'yustu/ as in I used to like it /ay'yustu'laykət/. The problem is whether the negative should be written