I see so many people here don't understand it... let me summarize it here (please correct me if something is wrong):
A. La cucina = The kitchen (femenine noun) / (then lE cucinE = The kitchenS)
B. Cucinare = to cook (verb)
And conjugated it is:
- io cucino = I cook
- tu cucini = You cook
- lui/lei CUCINA = s/he cooks (this conjugated form of the verb to cook happens to have the exact same spelling and pronunciation as the word kitchen, that's what might confuse you)
C. Un cuoco = A cook
I have no idea what you mean here. Remember the threading can be confusing so if you don't understand something be more clear in what you understand. Mediterranean's post at the top is quite good at clarifying the difference between 'la cucina' - the kitchen, 'lei cucina' - she cooks, and 'il cuoco' the cook.
I think "There is a grill in the kitchen" should have been okay, since it means essentially the same thing, but the software isn't that smart....So I think you're less likely to get it marked wrong if you stick to a more or less word for word translation, when that's possible.
It means a very similar thing, but when you're learning a language and the patterns of that language, and idiom isn't involved it's better to do as close a match as you can. There is a grill in the kitchen would be something like C'è una griglia nella cucina. Or possibly C'è una griglia in cucina (since using 'in' seems to obviate the need for the la).
Cucina can be a verb, as well as a noun.
The easiest way is to remember "la cucina" first as a noun (female), meaning "the kitchen/cuisine" and then as a verb in the third person form.
Verbs in Italian end with either -are, -ire or -ere, regular verbs drop this ending and replace it with 'o, i, #' for the 'I, you, he/she/it' form. The # is usually the vowel taken from the -#re, so the verb 'to cook' becomes '(io) cucino, (tu) cucini, (lui/lei) cucina' for 'I cook, you cook, he/she cooks'.
Other examples: Leggere: Leggo, leggi, legge. Mangiare: Mangio, mangi (no double i), mangia. Scrivere: Scrivo, scrivi, scrive.
EDIT: To incorporate Ariaflame's comment below and for clarification, the noun "cook/chef" is "cuoco" and not "cucina".