It seems like they all mean “in order to” or “due to”. From the description it appears that ため(に) and には are interchangeable and that には is just a contraction of ため(に). However is there an instance where one can be used and the other can’t?
Also, I am trying to reconcile the differences between のに and ため/には. There seems to be some nuance but I just can’t quite put my finger on it.
Finally, are these all colloquially used? Are there some that are used more in conversation.
Hey 
Basically,
のに is used to express that something is useful/important/needed/helpful in order to do something.
Basically, the importance of means to do something.
It is limited to expressions like 不可欠、必要、使う、要る、便利、欠かせない、必要不可欠 etc
Examples:
文法を学ぶのにBUNPROを使っています。
ご飯を食べるのに箸が必要だ。
マクドナルド を開設するのにお金は は欠かせない。
If you take a closer look at those expressions, you will notice that those can be also used with nouns marked by に by itself. 車が運転に必要だ。
ために is a more general expression, applying to all kinds of purposes. So basically you can always rephrase のに to ために but not always the other way around.
(Though if something that is beyond the speaker’s will is before ため then it means “because”, we can use “ように” so that instead)
ご飯を食べるために箸が必要だ。
マクドナルド を開設するためにお金は は欠かせない。
文法を学ぶためにBUNPROを使っています。
日本語能力試験を合格するために、学んでいます。
には
Is contraction of のには/ためには.
Well, that is basically all.
I hope it helps,
Cheers, 
Simple yet great explanation. Thanks for taking the time to do this write up.
Is there an example of when you can use ために, but のに doesn’t make sense?
「レースに出るために、車を買います。」is one example where ために cant be replaced with のに since it makes the sentence sound very unnatural.
When い adjectives and な adjectives are used のに is the correct word to use, but as mrnoone said, it can be replaced with ために.
When verbs are used, ために sounds more natural, but there are some cases where のにcould be used.
「レースに出るために、車を買います。」
The way I understand it, のに would be incorrect here because it would literally be impossible to race without buying a car.
However in a sentence like 「文法を学ぶのにBUNPROを使っています。」both can be used because BUNRPO is a useful tool for learning grammar, but there are other ways to do it.
Is that correct?
You make a good point! It could be incorrect because it would be impossible to race without buying a car. In the sentence 「文法を学ぶのにBUNPROを使っています。」, it does not really change the meaning of the sentence even if you used ために, but I think that’s a good way to look at it. It just gives it a slightly different translation.
「文法を学ぶためにBUNPROを使っています。」would translate as “I am using BUNPRO to learn grammar.” 「文法を学ぶのにBUNPROを使っています。」would translate better as “In order to learn grammar, I am using BUNPRO.”
However, speaking from a native perspective, using のに in the first example just sounds wrong. It is similar to when you hear a nonnative English speaker use a grammar point in their sentence and sounds wrong. You understand what they are saying, and the sentence makes sense, but it just sounds “wrong”. I am sure many people know to use ために after verbs and のに when adjectives are used, but I think most people just go with what they think is “right” or sounds “natural”.