There's a subtle difference. Adding «есть» will emphasise the existence of plans, while dropping it shifts emphasis to the word 'plans' itself. I believe both work here. In some other contexts, «есть» is neccessary.
Hm, that's a difficult question. Sometimes sentences with «есть» sound better, sometimes sentences without it.
I would suggest this: if you just have a noun without an adjective or other modifiers, then donʼt drop «есть»:
- В ко́мнате есть стол. (There's a table in the room.) — the most important piece of information is that there's a table in your room
- У меня́ есть ноутбу́к. (I have a laptop) — the most important piece of information is the existence of a laptop
But if you have an adjective, a numeral, or other modifiers, then drop «есть»:
- В мое́й ко́мнате но́вый стол. (There's a new table in my room) - the most important piece of information is that the table is new
- У меня́ ноутбу́к с Linuxʼом. (I have a Linux laptop) — the most important piece is that there's Linux on your laptop
- У Ма́ши две подру́ги. (Masha has two [female] friends) — the most important fact is the number 2
This is not a rule (because the sentence «У меня планы» would never be produced if you followed my suggestions), but you can use this as a shortcut until you understand the pattern better.
And that's exactly why I did not write THE plans. But I was marked wrong. In English, if I use THE, the existence is already assumed; with-out it, it is indicated in the sentence. Am I misunderstanding some-thing, or is Duolingo messing up? Another ex. in this lesson goes exactly the other way round. I am really confused.
You would use genitive for both 'our' and 'plans', i.e. «содержа́ние на́ших пла́нов».
Yes, «наш» 'our' behaves like adjective and matches the modified word in case, number and gender. The same is true about мой 'my', твой 'your' (informal singular), ваш 'your' (polite or plural).
Его, её and их, however, work differently.
«У» is a preposition, here it means something like ‘at the possession of’. (Its original meaning is ‘near’.)
So, basically, the sentence structure is ‘at_the_possession_of us, [there are] plans’.