Translation:Please give me a thick pen.
Yea, what he said is just that in plain form using だ after an adjective is wrong, but です is correct in polite form.
I can add that japanese adjectives are coniugated just like verbs. Duolingo never explained this, but there are -い and -な adjectives. The former can be made negative/past by dropping the い and adding -くない / -かった; while latter just drop the な and add だ (です in polite) and coniugate as usual.
そのセリフは汚かった(です) That phrase was dirty
あの公園はきれいだった That park was beautiful
In other words -い adjectives are "true" adj and they can be used alone as the predicate. -な adj are actually just nouns used as adjectives, hence they are not considered "true" and they need だ to work as predicate.
I'm having trouble and getting a lot wrong, like this one, not because I don't know the Japanese, but because I use different English. For example, I used the word jumper instead of sweater, the right hand one instead of on the right, I put the cup into the box instead of in the box, and here, could I have instead of give me.
In the case of British English (jumper instead of sweater), just report it, you're right. The Japanese course is still in beta, so not all of the British options have been added yet. Because I don't know the specific translations that you're referring to, there may be a reason that your answers were wrong, but it was most likely just a case of your answers not being added to the options yet.
For this particular translation, the word "kudasai" means "please give me". It's a fairly polite request, but not as polite as "could I have". It's not a word that you would typically use outside of your in-group. In my opinion "please give me" is the best and standard translation for "kudasai".
For showing the computer that you understand the word ください, in my opinion it's wrong. If you were working as a professional translator and had to consider the context of the sentence using ください and the audience that you're localizing the translation for, I think it could be correct to interpret it as "could I have". You can always submit an error report and see if the contributors will accept it or not.