It's not «картофел» but «карто́фель». It's indeed borrowed from German. «Карто́шка» was originally a diminutive form of «карто́фель».
Indeed, there are two similar words: есть 'to eat' and есть '(there) is/are/am'.
Есть 'to eat' is an infinitive. It's used when the sentence has a different main verb, for example: я хочу́ есть 'I want to eat'. Here, хочу́ 'want' is the main verb, so е́сть is used in its infinitive form.
\1. When е́сть 'to eat' is used as the main verb, you use its personal forms. There are 6 of them:
- я ем 'I eat', 1st person singular,
- ты ешь 'you eat', 2nd person singular (this form is familiar, used with friends),
- она́ ест 'she eats', 3rd person singular (this form is used with all the singular nouns: котёнок ест 'a kitten eats', соба́ка ест 'the dog eats', etc.);
- мы еди́м 'we eat', 1st person plural,
- вы еди́те 'you eat', 2nd person plural (it's also used instead of singular for politeness),
- они́ едя́т 'they eat', 3rd person plural (this form is used with all the plural nouns: ко́шки едя́т 'cats eat', щенки́ едя́т 'puppies eat').
\2. When есть means 'is/are/am', it's a form of the verb быть 'to be'. However, быть is a tricky verb. In present tense it's often omited, e.g. «Я челове́к» 'I am human', «Я жа́ба» 'I am a toad'.
It's not omited in sentences that are 'there is': «Возле пруда́ есть жа́бы» 'Near the pond, there are toads'. But when it's not omited, its peronal forms are unlike any other verb:
- я есть 'I am',
- ты есть 'you are',
- она́ есть 'she is',
- мы есть 'we are' (rarely, in very bookish texts: мы суть),
- вы есть 'you are',
- они́ есть 'they are'.
Basically, it doesn't change at all!
Sentences about 'having' are in fact sentences about 'being' in Russian:
- У меня́ есть жа́ба 'I have a toad' (literally: 'At my [possession], there is a toad'.)
- У ма́мы есть э́та кни́га. 'Mum has this book' (literally: 'At Mum's [possession], there is this book.')
So, basically, sentence about 'having' are sentences about 'being' in Russian.
\3. You can distinguish the two usages of есть because they are not used in the same sentences:
- есть 'to eat' is in infinitive, so it is used with another verb;
- есть '(there) is' is a personal form of the verb, so it's not used with another personal form of the verb.
So, if a sentence has another main verb, then «есть» is an infinitive and it means 'to eat':
- Я хочу́ есть. 'I want to eat.' (хочу́ 'want' is the main verb)
- Я не успе́л пое́сть. 'I didn't have time to eat.' (успе́л 'had time' is the main verb)
When есть is used as the main verb, it means '(there) is/are/am':
- У меня́ есть компью́тер. 'I have a computer' (literally: at my [possession], [there] is [a] computer)
- нас = genitive of мы (у is used with Genitive to show the possessor),
- картошка = nominative singular (it's a mass noun, so we don't make it plural).
I think that your sentence is talking about a specific (but uncounted) number of potatoes, so you'd have to use some Russian plural form of "potatoes". What I've read in the comments suggests that you want to use картофеля (nominative masculine plural - the singular nominative masculine form is картофель) rather than картошек (nominative feminine plural of картошка).
From other comments, it seems like you're supposed to use the German-based картофель (nominative masculine singular) for a single potato, although context is important. I wish I knew what that context is, because картошка is singular but translated as plural because it's a mass/collective noun.
Карто́фель is also a collective noun, too.
Singular is карто́фелина (or карто́шина, but that’s less commonly used).
«Наш» is an adjective-like pronoun that is usually followed by a noun. «Нас» is a form of «мы» (genitive case form) that is used after a preposition «у».
«Есть» (because the quesion is ‘do we have potatoes or don’t we have potatoes’? in fact the whole group «есть картошка» would be emphasised but «есть» would be more prominent), but the emphasis should be relatively small (it’s a sentence with the neutral word order so the emphasised word gets much less emphasis than in sentences with non-standard word orders).