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밥이 국에서 먹어도 됩니까? This is a grammatically correct sentence but its concept illogical. Literally, it means: Can rice eat in(side) the soup?

May I eat the rice in the soup? 밥을 국에 말아서 먹어도 됩니까? sounds best.

너무 추워요. 잠바를 입어도 돼요? 잠바 tends to be a jacket rather than a jumper as the word suggests. The word for sweater is 스웨터.

It's really cold. Can I wear a sweater?

오늘은 직업을 바쁘지 않입니다. 일찍 가도 됩니다. 오늘은 일이 바쁘지 않습니다. 
- 을 is a determiner that is used with a noun that expresses prediction, will, determination or ability, and is not an established fact. E.g. 나는 오늘 안에 밀린 일을 다 해 놓을 것이다. ('해 놓을' to complete) I am going to have caught up on all errands today.

-을  The most common uses of this as a postposition is when the noun is the object, e.g. 밥을 먹다.

지 않- and  -습 is the correct combination here.

-입니다 is used after a noun.

Work isn't busy today. You may leave early.

2) 면 안 되다.

직업에서 공부면 안 됩니다. 일하면서 공부하면 안 됩니다. This sentence would mean 'You cannot study and work at the same time.' 공부하다 is used as a verb here. If you want to use it as a noun, you could try: 공부를 직장에서 하면 안 됩니다. ('직장' workplace)

Studying at work is not allowed.

차로 운전하면서 먹으면 안 됩니다. ✓

Eating is not allowed while driving your car.

과자를 영화관에 가져오면 안 돼요. ✓

You're not allowed to bring snacks to the movie theater.

3) -지 않아도 되다

공원에 오지않아도 돼요. Put a space between 오지 and  않아도.

You don't have to come to the park.

치킨을 주문할 거예요, 점심을 만들지 않아도 돼요. ✓

I'll order chicken, you don't have to make lunch.

무게 얻었어요, 제가 후식를 먹지 않아도 돼요. You can use a full stop instead of a comma.

Yes, you can say 무게 얻었어요. It sounds unnatural, though, so 살 졌어요 is correct.

제가 or 저는? Which one fits in here? -가  gives emphasis to 'I' -는 compares one with another. 저는 후식을 먹지 않아도 돼요.
I gained weight, I don't have to eat dessert.

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-> To be honest, I don't know how to translate it right. So, I'll show some possible examples.

밥과 국을 먹어도 됩니까?

국 안에 있는 밥을 먹어도 됩니까?

밥을 국에 말아서[넣어서] 먹어도 됩니까?

cf) 국soup 찌개stew 죽congee

너무 추워요. 스웨터를 입어도 돼요?

It's really cold. Can I wear a sweater?

-> 스웨터sweater, 잠바(not correct but prevalent)/점퍼jumper

오늘은 일이 바쁘지 않습니다. 일찍 가도 됩니다.

Work isn't busy today. You may leave early.

-> Actually, so many Korean use 오늘은, but to avoid the misunderstanding that you could understand it 'today is~ 오늘은' as like a subjective particle, I'd love to recommend to use 오늘, not 오늘은.

2) 면 안 되다.

직장에서/일 하면서 공부하면 안 됩니다.

Studying at work is not allowed.

-> at work 일 하면서while working  or 직장에서 at your company

차로 운전하면서 먹으면 안 됩니다.

Eating is not allowed while driving your car.

과자를 영화관에 가져오면 안 돼요.

You're not allowed to bring snacks to the movie theater.

3) -지 않아도 되다

공원에 오지V않아도 돼요.

You don't have to come to the park.

치킨을 주문할 거예요, 점심을 만들지 않아도 돼요.

I'll order chicken, you don't have to make lunch.

무게 얻었어요,살 쪘어요. 저 후식를 먹지 않아도 돼요.

I gained weight, I don't have to eat dessert.

-> gain weight 살이 찌다