I've a visa-free passport to UK but was refused to enter in Heathrow. Applied for a visa as advised by IO but still refused

You are a Malasian national and accordingly do not require entry clearance to visit the UK. Earlier this year you were removed from Terminal 5 LHR under Paragraph 320. You then applied for entry clearance and were refused under Appendix V 4.2 (a) & (c). It's all bad, but most damningly, the Immigration Officer made a good case that you intended to defraud the NHS. This is the type of thing that gets them very upset, it echos the British public mood of frustration at people abusing the NHS and you can expect the authorities to stay upset about this for a long time.

You have a lot of questions about these events, too many for us, but you raise a few points not already covered in our archives here that are valuable. Let's have a look at them...

If my appeal for my Bar exam gets approved, and in my application I mention that I'm visiting the UK to meet my fellow classmates and lecturers to get materials for my exam and also sit for the exam in March 2017 (it's actually March not April 2017 as I've checked the timetable) would that consider a change in circumstances?

No. This highlights a big misconception that people with a history of refusals often have. You want to change the premise of your application. That's fine, but changing the premise is not a change of personal circumstances. They are two different things. Every time somebody applies at an airport or using visa4UK, they give a premise. A new premise does not change the underlying facts about YOU, i.e., your lifestyle, your social ties, your economic situation, and so on. If your circumstances are all the same, then you will not qualify and can expect a refusal.

Another interesting point you raise...

My partner just started her work and is working with a working agency. Her mom is still supporting her. Should I also attach her mum's employment and financial status?

This is a bad idea; it's grasping at straws. If you are in reduced circumstances, and your partner is in reduced circumstances, and your partner's mum is in reduced circumstances by having to support two adults, then no, this scenario does not depict a genuine visitor.

How can I be sure after fulfilling both notice of refusal(the one I received at the airport and the other of my visa application) points that they will accept my application and that they wont give me new set of reasons of refusal?

The remedy for this is to change your personal circumstances such that you are an established professional on a stable career track in Malaysia with a great history of solvency and financial independence. This is what they want to see and the best recipe for success with UKVI. In particular, they love to see lots of travel history to the USA or Canada or Australia; they eat it up and it's a wonderful calling card in a visa application.

How can I get out from the shadow of the edited and omitted documents that I brought?

The normal advice for this is to instruct a UK solicitor with a practice area in removals to represent your next application. They can think up polished and eloquent explanations (that are truthful and not lies) that Entry Clearance Officers can appreciate. It's how they make their living and what they are paid to do. Given that your reserves are about GBP 3,500, this isn't an option for you. So the advice above about developing an independent lifestyle applies.

I think that would make sense, as they know that I need to still come back to Malaysia and sit for my exam. My concern is that I just wanna see my partner (she's a British citizen) and go back that's it.

You have missed the point. Overstaying is the smaller worry they have. The damning issue is that they have it in mind that you intend to break our laws whilst in the UK by defrauding the NHS. And they made a good case for it also. Trying to wiggle off the hook by saying it's for someone else does not exclude you from participating in a joint enterprise.

Incredible to relate, you do not have a ban so Paragraph 23A of the rules kick in. It means Malaysian nationals do not require prior entry clearance and you are free to book an air ticket and try your luck in Heathrow all over again. Alternatively you can apply again from Malaysia for entry clearance and avoid the distress of wasted airfares and removal from port. The outlook is not favourable for either of those alternatives owing to the things pointed out above, but it's your call.

All of the various paragraphs and appendices cited in this answer are at the Home Office's site: Immigration Rules - Guidance.

The removal notice you got in LHR has your case number on the top, hang on to it.