I understand your frustration but the way you worded your emails you are not going to get any real help from IHG Care which has never been a customer-friendly entity to begin with.

Treat others as you want to be treated would get you much farther.

Also Firstly and Secondly are correct usage of adverbs in the context.

I were you I would write this way after they sent you a boiler plate of waiting 3 weeks for the miles to post: Yes I understand it takes time for the miles to be sent to the partner for posting. That is why I have waited 10 weeks to give ample time for it to be processed. Since there is still no mile shows up after 10 weeks have passed, could you please open an investigation to verify that miles are indeed sent? I appreciate your help to set up a case file on this and look forward to hear from you for a resolution as soon as possible.

The point is, you need their help to get your miles. So you plead your case politely and prod them to take action.

You can argue that it is their job, they should do it regardless. However a more tactful approach would probably get what you want sooner than never.