achim tellebijeon nyuseureul bomyeonseo babeul meogeosseo.

먹었어 is the banmal (informal) past tense of 먹다. To make it polite (jondaetmal), change it to 먹었어요. The first verb’s connective form stays the same, so you’d say:
• Informal: 아침 텔레비전 뉴스를 보면서 밥을 먹었어.
• Polite: 아침 텔레비전 뉴스를 보면서 밥을 먹었어요.

Here, 아침 is part of the compound noun 아침 텔레비전 뉴스 (“morning TV news”), so no particle is needed. If you want to say “in the morning” as an adverbial time phrase, you use 아침에 and treat 텔레비전 뉴스를 separately:
아침에 텔레비전 뉴스를 보면서 밥을 먹었어.

In Korean, subjects (and objects) are often dropped when they’re clear from context. Here the implied subject is “I” (나/저). If you need to emphasize it, you can say:
나는 아침 텔레비전 뉴스를 보면서 밥을 먹었어.

Yes. Both sentences describe two simultaneous actions, but the main clause (the part not in -면서) feels more in focus.
• 보면서 밥을 먹었어 → Focus on eating while watching.
• 밥을 먹으면서 뉴스를 봤어 → Focus on watching while eating.

• -면서 specifically stresses that the actions overlap exactly (“while doing X, Y”).
• -고 simply lists or sequences actions (“X and then Y” or sometimes simultaneously, but less emphatic).
Example:
– 뉴스를 보면서 밥을 먹었어. (I was eating breakfast while watching the news.)
– 뉴스를 보고 밥을 먹었어. (I watched the news and ate breakfast—less clear if at the same exact moment.)

For most Korean verbs, you drop -다 and add -으면서 or -면서 (if the stem ends in a vowel). 보다 → stem 보 → 보면서.
You use -하면서 with 하다 verbs or Sino-Korean nouns:
• 공부하다 → 공부하면서
• 청소하다 → 청소하면서

When a verb takes an object, you normally mark it with -을/를:
• 뉴스를 보다
• 밥을 먹다
In very casual spoken Korean, particles can be dropped if the meaning is obvious:
• 뉴스 보면서 밥 먹었어.
But in writing or more careful speech, it’s best to include -을/를 for clarity.