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Hi, I am thinking to apply for UK residence card as EEA unmarried partner.
Me (from Taiwan, tier 5 visa expire in March 2021)
My boyfriend (EEA, staying in the UK for over 10 years)

We have been 3 years together, but didn't live together for 2 years. Our priority is to apply for residence card, but if our situation seems hard to pass the application, we would consider getting married before end of this year.
1. The period we live together: Oct 2017-Jan 2018 (4 months), Aug 2019-Now(8 months)
We met when I was studying for master degree in the UK. After my student visa expired, I went back to Taiwan for work. We spent 1.5 years in long-distance relationship. And I came back to the UK because I was granted Tier 5 temporary working visa.
2. The proof we have: my bank statement(12 months) and payslip(5 months) sent to his flat, our traveling pictures(including 3 abroad), trips during long-distance relationship(we met every 3 months), cards through international mail, pictures with our family
3. Financial condition: He has been working in the UK for 10 years, and the salary meets the requirement. I am working in the UK for 5 months so far, and my salary is above the minimum wage.

What else do we need to prove our durable relationship?
If we fail to gain the residence card, would we face any trouble to get married straight after?
What are my chances to obtain residence card based on our circumstances?

We have been 3 years together, but didn't live together for 2 years.

A very common mistake that people considering the unmarried partner route make is to interpret that the main requirement is to simply cohabit for 2 years.

That is an incorrect interpretation. The main requirement is to have functioned in a similar way as a married couple for two years, and this, in the UK, translates as having joint financial commitments for two years.

For example, you can cohabit with someone for 10 years and be in a relationship with them, but if you lead substantially independent financial lives, and do not have joint strong financial commitments (e.g. a joint bank account, a mortgage, the joint responsibility of a child) then you simply are not considered a durable partner, for the purpose of this immigration route. This does not mean your relationship is not genuine, it simply means your relationship is a boyfriend/girlfriend relationship that is not considered to be "akin to marriage". A relationship like this, does not qualify for the unmarried partner route.

if our situation seems hard to pass the application, we would consider getting married before end of this year.

Always a valid option.

1. The period we live together: Oct 2017-Jan 2018 (4 months), Aug 2019-Now(8 months)
We met when I was studying for master degree in the UK. After my student visa expired, I went back to Taiwan for work. We spent 1.5 years in long-distance relationship. And I came back to the UK because I was granted Tier 5 temporary working visa.

Very likely not enough if not accompanied with strong proof of joint financial commitments.

2. The proof we have: my bank statement(12 months) and payslip(5 months) sent to his flat, our traveling pictures(including 3 abroad), trips during long-distance relationship(we met every 3 months), cards through international mail, pictures with our family

I'm afraid this only vaguely proves you cohabited for a non-continuous period of less than 1.5 years. As explained above, not enough.

3. Financial condition: He has been working in the UK for 10 years, and the salary meets the requirement. I am working in the UK for 5 months so far, and my salary is above the minimum wage.

Completely irrelevant. For the EEA route, there is no salary requirement for either of the parties, you must be confusing this with the route for family members of UK citizens. Again, the main financial requirement is joint commitments, which sadly seems that you do not have.

What else do we need to prove our durable relationship?

Having read all my comments above, it is up to you to think if your relationship is durable or not. Again, the question is not if it is genuine, the question is if it's durable and "akin to marriage". From a personal perspective, and according to the information you have shared, it seems your relaitonship is genuine, but not durable.

If we fail to gain the residence card, would we face any trouble to get married straight after?

it is not unreasonable to think it will look better if you do not have a refused residence card application on your record when submitting your intention of marriage,

What are my chances to obtain residence card based on our circumstances?

All applications are different and considered on a case by case basis, but purely based on the information that you have shared, it is not unreasonably to think that your chances of getting an unmarried partner residence card are Very low.

Reading your circumstances, it seems to me like you have a weak application here.

Even married partners are sometimes challenged about their relationship. But usually they have strong response elements to that. In your case, you seem in a relationship where you are on/off cohabiting. During these times, you send him your part of the rental. But this is not how it works to show joint financial commitments.

You need to show a joint account, both names on bills such as council, utilities... etc.

These joint financial commitments are stronger and harder evidence then showing that you went on holiday together.

In the light of what you shared, I think the chances of refusal are much higher than chances of positive outcome.