The phrase in the second sentence 'for the purpose that' doesn't really exist in English.
For the third sentence, 'in order that' is rarely used in my opinion, and it is extremely formal. So your last sentence works, but you very very rarely see it.
With 'for the purpose of' you can follow this phrase with an 'ing' form or a normal noun:
'The book is for the purpose of studying English' - ing form
'The book is for the purpose of English practice' - normal noun
The third option - pronoun + 'ing' form is never really used. This is possibly because 'for the purpose of' is too long as a phrase, and it doesn't really flow.
In both of these cases, just using the word 'for' is probably better:
'The book is for studying English'
'The book is for English practice'
With 'in spite of', you can use:
A noun - 'In spite of the rain we continued playing football'
An 'ing' form - 'In spite of losing we were very happy'
A Prounoun/noun + 'ing' form - 'In spite of them winning we were very happy'
With many verbs/nouns/ adjectives + prepositions, you can follow them with all of these. For example if we take 'to agree with something':
'I don't agree with this plan'
'I don't agree with smoking'
'I don't agree with him smoking'
With 'in order to' you have to follow this with an infinitive. Also with 'in order to' you must have a full clause (sentence) before it with a subject object and verb.
This means that with 'in order to' you can't just say: 'The book is in order to study English.'
For this to work you must have a full subject - verb - object:
'I have bought the book in order to study English'
'In order to' is also quite a long phrase, which means it is also quite formal and can slow down the flow of the sentence. To replace this we usually just use 'to':
'I have bought this book to study English'
Hope this helps, let me know if you have any questions!