Whether-If – Can ‘Whether’ Replace ‘If’ in This Sentence?

SHORT ANSWER:
You are exactly right; (D) is not usually acceptable^†, but all the others are.

There are however other circumstances in which whether and if are used somewhat differently, and different rules apply.

LONGER ANSWER:
You raise, in effect, three distinct questions:

1. When may if be substituted for whether?
   This is an easy one: if may always be used in any expression which employs whether. (But as we will see, some syntactical adjustments may be required.)

2. When is or not permitted or required in expressions with whether and if in that sense, and how is it deployed?
   Whether and if are used to introduce clauses expressing a choice between two or more alternatives in two contexts:

   • In complement clauses

     I need to know whether this word fits in this sentence or does not fit in this sentence or fits only sometimes.
     I need to know if this word fits in this sentence or does not fit in this sentence or fits only sometimes.

     I need to know whether this word fits in this sentence or does not fit in this sentence.
     I need to know if this word fits in this sentence or does not fit in this sentence.

     In situations like the second pair, where there are only two alternatives and one is simply the negation of the other, you may replace the second alternative with a bare not:

     I need to know whether this word fits in this sentence or not.
     I need to know if this word fits in this sentence or not.

     In fact, you can omit the or clause altogether; the absence of an explicit or implies that the negative is intended:

     I need to know whether this word fits in this sentence.
     I need to know if this word fits in this sentence.

     The phrase or not may be moved immediately after whether, but this is not generally used^† with if:

     ^okI need to know whether or not this word fits in this sentence.
     ^??I need to know if or not this word fits in this sentence.

   • in condition clauses
     When whether and if are used to head the ‘protasis’ (IF or condition clause) in a conditional construction, the rules are different.

     With whether, there must be an explicit or clause or phrase:

     ^okWhether the word fits or does not fit, spell it correctly!
     ^okWhether the word fits or not, spell it correctly!
     ^okWhether or not the word fits, spell it correctly! ... BUT
     ^∗ Whether the word fits, spell it correctly!

     And if cannot be simply substituted for whether in a condition clause, because in that context whether implies that the consequence is true or actualized if either condition is in force. You have to employ an if clause that expresses that meaning. With just two alternatives you may employ either or if or both if...and if:

     If the word fits or if it does not, spell it correctly!
     Both if the word fits and if it does not, spell it correctly!

     These are grammatically and logically sound, but clunky; and there really is no graceful way to express three or more alternatives using if, since the introductory both, which alerts the reader to the structure which will follow, is not applicable. Stick with whether.

3. Are these uses of if restricted to informal use?

   It's not really a matter of formality v. informality, but of monologue v. dialogue. In conversation, where any misunderstanding is easily corrected, if is entirely acceptable.

   But in writing, or in spoken contexts such as lectures and broadcasts which do not involve participation by the hearer, you want to make what you are saying as easy for the addressee to ‘process’ as possible. The “rule” here is to use the most explicit term possible. Since if has a very broad range of syntactic and semantic meanings, while whether has a relatively narrow range, your choice should always be to use whether rather than if in circumstances where either is acceptable.

   And there is really no reason to avoid following this practice in conversation as well. It will not mark your speech as pedantic or highfalutin—everybody understands and uses whether.
   
   

   ---

^†FumbleFingers points out that this use does appear from time to time, but acknowledges that it sounds “strange”. That, to my mind, means ‘Don’t use it’.

^∗ marks an utterance as ungrammatical.

It seems that "doubt whether" was far more common for a long time, but the usages have largely converged, and both appear to have currency.

Personally, I prefer "whether" in both of these cases, but I would not see a big distinction between them. Neither usage would mark a speaker as being non-native or particularly regional to me, nor seem out of place. I would find "if" in these sorts of sentences to be somewhat more casual, however.

I am not aware of any rule governing which one to use... if you're in doubt, I would go with "whether" when you are discussing a true-or-false/yes-or-no condition in an object clause like this.

You didn't ask, but note that "I doubt they'll make it" is also acceptable. That's particular to the verb doubt though.