The keyword is ‘ㅣ’ 모음 순행 동화 by which ㅓ and ㅗ following ㅣ becomes ㅕ and ㅛ, respectively.
According to 표준 발음법, 기억 is not allowed to be pronounced as 기역 (기억 is the correct pronunciation) because 억 is not an ending of a verb or adjective; however, some people go against the rules. I would say that it is natural to pronounce 기억 as 기역 especially when 기억 is quickly read. 기역 that is the name of ㄱ is rarely mentioned in daily conversation, so the sound 기역 will be interpreted in Koreans' brains as 기억.
As for 용언 (verbs and adjectives) such as the verb 피어나다, ‘ㅣ’ 모음 순행 동화 is allowed for their 어미 (endings) by 표준 발음법:
다음과 같은 용언의 어미는 [어]로 발음함을 원칙으로 하되, [여]로 발음함도 허용한다.
피어나다 is a compound verb (피 + -어 + 나다) and -어 is an ending. Therefore, 피어나다 can be pronounced as 피어나다 or 피여나다. You should note that 피어나다 and 피여나다 sound different. The sounds of 피어나다 and 피여나다 can be found here.
좋아하다 is irrelevant to the keyword that I mentioned above. 조ː아하다 (not 조와하다) is the correct pronunciation because the final consonant ㅎ that is followed by a vowel is silent. This rule is more clearly specified in 표준 발음법:
‘ㅎ(ㄶ, ㅀ)’ 뒤에 모음으로 시작된 어미나 접미사가 결합되는 경우에는, ‘ㅎ’을 발음하지 않는다.
"連音音變" seems to be Chinese. 連音 (连音) means linking; 音變 (音变) means 음운 변화 (phonological change).
Under normal conditions, it is possible to distinguish between 조아 and 조와.
To understand how different ㅗ와 and ㅗ아 sound, you should listen to the pronunciation of 보완 and 보안 (You may use the standard dictionary instead). These 완 and 안 sound clearly different to me.
It is wrong to think that every Korean character represents its actual sound. Modern Korean ㅘ is read as ᆉ (ㅜㅏ) because it includes the /w/ sound. Even this learning material mentions ㅜㅏ. Therefore, even if you read 오아 quickly, it is still 오아 not 와. A document also says ㅘ starts with the sound ㅜ:
- One more thing I need to say is that some people are too lazy to focus on the starting sound of ㅘ (Then, it is just ㅏ). For example, they pronounce 과일 as 가일; however, 가일 is the incorrect pronunciation. This tendency is also mentioned in a Korean book. They might read out 조와 as 조아, but it can be a hasty generalization.