질투의 화신 14회 (Learning thru Kdramas)
Can somebody clarify some of the scenes?

1. Hwa Shin asked Na Ri about numbers with long vowels. These are 이, 사, 오, 만, 두, 세, 네, 쉰. (Although subtitle only mentioned 2, 4, 5 and 50). Then rest are all short vowels. Although the way I understood that scene is that those numbers have one syllable only so she needs to pay attention on those numbers.

2. Alternate usage of Native and Sino-Korean numbers in news.
54살 조모씨-Native
54세 김모씨-Sino
69살 최모씨-Native
24세 이모씨-Sino
55살 정모씨-Native
24명-Native
1228건을 – 천이백스물여덟 (Is it a mixture of Native and Sino-Korean numbers?)

3. On the 18th minute up to the 20th, it seems like a tongue twister. Do ordinary Koreans could be able to speak that fast without mistakes?

4. Hwa Shin was called by his mom ‘baby’. The dictionary form of baby is아기. Why his mother mentioned him 애기and 새끼? Later on, Ms. Gye and Ms. Bang called Hwa Shin ‘애기야’, ‘아가’. Are those terms considered slang?

Question

질투의 화신 14회 (Learning thru Kdramas)
Can somebody clarify some of the scenes?

1. Hwa Shin asked Na Ri about numbers with long vowels. These are 이, 사, 오, 만, 두, 세, 네, 쉰. (Although subtitle only mentioned 2, 4, 5 and 50). Then rest are all short vowels. Although the way I understood that scene is that those numbers have one syllable only so she needs to pay attention on those numbers.

2. Alternate usage of Native and Sino-Korean numbers in news.
54살 조모씨-Native
54세 김모씨-Sino
69살 최모씨-Native
24세 이모씨-Sino
55살 정모씨-Native
24명-Native
1228건을 – 천이백스물여덟 (Is it a mixture of Native and Sino-Korean numbers?)

3. On the 18th minute up to the 20th, it seems like a tongue twister. Do ordinary Koreans could be able to speak that fast without mistakes?

4. Hwa Shin was called by his mom ‘baby’. The dictionary form of baby is아기. Why his mother mentioned him 애기and 새끼? Later on, Ms. Gye and Ms. Bang called Hwa Shin ‘애기야’, ‘아가’. Are those terms considered slang?

Answer

1.
Although people usually tends to ignore, there is long vowel word and short vowel word in Korea.
For example,
눈(short) means "eye"
눈[눈ː](long) means "snow"
밤(short) means "night"
밤[밤ː](long) means "chestnut"
이(short) means "this"
이[이ː](long) means "two"

Likewise, 이(2), 사(4), 오(5), 만(10000), 두(2), 세(3), 네(4), 쉰(50) in numberal have long vowel sound.
You can see the symbol [~ː] in naver dictionary and any plublised Korean-Korean dictionary.

2. 
It depends on the counting noun.
Generally,
if the counting noun is Sino-Korean, the numbers read as Sino-Korean.
if the counting noun is Native-Korean, the numbers read as native-Korean.
There are some exceptions.
건, 개, 명, 시간, 근, 시 are used with native-Korean number

살 is native, 54(살) read as native.
세 is Sino-Korean, 54(세) reads as Sino-Korean.

When reading as Native number, the number which is above 100 read as Sino and the number which is below 100 read as native.

3.
It is not easy for ordinary people to speak fast those a tongue twister.
It requires a lot of practice to speak that fast without mistakes.

4.
All those terms are not slang. They are terms used for informal conversations among intimate family members.