I don't think bento needs to be, or rather, should be translated. Bento itself is a very specific thing, that doesn't really exist anywhere else and most people - especially people taking a Japanese language course - are going to know what a bento is. It simply doesn't need to be translated. Moreover, where I'm from, if you asked me if I wanted a lunch box, I would assume you are asking me if I wanted a container to put my lunch in.

I feel the same way about お姉さん and お兄さん those two terms are something intrinsic to Japanese society, most people will know (or very quickly figure out) what it means, and it just sounds incredibly awkward when translated. Most good translators don't bother translating it, and it's easy to see why - I have a brother, and not once in my entire life did I ever call him 'big bro'.

For 2, おべんとうはどうですか

Yes, see AndreeCora's comment above.

Not that Japanese would appreciate it.

But if I came across a Japanese person eating bento and I have the same status at least, I would say dou?

I think that would be a valid translation.

I don't think saying "lunch box" would be the best translation, since a bento is a very specific thing, not really the same as just your average lunch box from say, America, or some other place.

I wrote "how is the bento" and it was accepted

Using "the" sounds like you are asking about the lunchbox's feelings.

That kind of how is どう

"Ikaga" is usually written in hiragana.

Like is it the same thing? Is it a sentence that depends on context?

They might be able to infer the same meaning but there is a distinct difference. One is offering, the other is enquiring about desire... If you can differentiate with two English sentences then you can obviously differentiate between two Japanese sentences. Think about literal meaning behind words you learn. Once you understand those then you can start speaking colloquially.

Sounds correct to my ear, but maybe "how about a bento?"

I wanted to ask the same. For me it seems to be better than "Would you like a bento?"

I feel like "how about that bento" is the closest English equivalent that covers both meanings.

It really means "How about" or "How would you like". It's always phrased as a suggestion, as in something that's being offered. The person saying this is suggesting you might want X. it's a common phrase to hear when shopping.

Bento is a single item, so it would never be "some bento"

Because a bento is a bento, in the same way that sushi is sushi. There is no translation for it. In English it's also a bento, or a bento box. In Hawaii, for example, it's quite common for grocery stores to have "bento boxes" and that's what they're called in English.

Not unless something or someone else were the topic of the sentence. "Ha/wa" indicates that "bentoo" is what the sentence is focused on. "Ga" just indicates the relationship with the verb in the clause.