This is a very good question. Why has no one answered you for so long? I'm not an expert, but I'll try my best to help you.
I'm only speaking from feeling here, so I could be wrong on some points. Consult a dictionary if you're unsure.
1/ type , sort , kind , category
These words are all synonyms. You can use type, sort, and kind interchangeably in most cases. There are only very slight differences in their meanings:
"kind" and "type" are nearly identical. But sometimes they do differ. Saying "I don't like your kind" and "I don't like your type" have different meanings. "Your kind" refers to race. "Your type" refers to appearance or personality.
"Sort" has a somewhat negative meaning to it. If you begin a sentence with "This sort of <something> is ...", it is almost always a negative remark.
"Category" is often used when you are talking about objects or topics that have been formally categorized.
2/ allow , permit , agree
"Allow" and "permit" have the same meaning, but "permit" is used in written language. You'd rarely hear this word used in spoken conversations. "Agree" has a different meaning from the other two. Please refer to a dictionary.
3/ postpone , delay , put off , cancel
"Postpone" is intentionally done. "Delay" is often unintentional.
"Put off" is identical to postpone, but is often used with events that you'd rather not do. "Cancel" has a different meaning. It is to abort the activity altogether.
4/ cure , heal , treat
"Cure" is when you remove an illness entirely. "Heal" is often used with wounds. It means the wound is getting better. "Treat" is the mere act of attending to an illness or a wound.
5/ provide , supply , offer
To "provide" is to give. "Supply" is long-term giving. "Offer" is to the act of asking something if he/she would like to be given something. It's not the act of giving.
6/ occur , happen , take place
"Occur" is often used with events that you know will happen in advance such as eclipses. "Happen" is often used with something you don't know in advance, like an accident. "Take place" is often followed by "at" to tell the location of the event. These are not strict rules. All three can be used interchangeably.
7/ opinion , idea , view
"Opinion" is what you think of something (good, bad, like, dislike, etc.). "Idea" is a thought that you have relating to a topic, with or without opinion. A "view" is similar to an opinion.
Hope this helps. I suggest finding an English-English dictionary with sentence examples to help you understand the words better.