Translation:I do not want to go to the hospital.
The grammar is at http://www.guidetojapanese.org/learn/grammar/desire.
Basically, just conjugate the verb's ‐i stem by appending ‐tai and then the verb's base meaning changes to "want to do (verb)" and gains the grammatical properties of an ‐i adjective.
Change the ‐i at the end of ‐tai (like one would do for an adjective) to ‐ku and you've changed the verb into an adverb.
The result is "There isn't want to go to hospital." in pseudo‐English.
I think he may have been thinking about how in English we often put ourselves mentally in the location of the person we're speaking to. On the phone you might say, "I want to come to the party" if a friend has just told you they are going to one. As far as I know, this doesn't happen in Japanese so you always speak with regard to your own location. So in the previous situation, in Japanese you would say パーティーに行きたい。
Not in this case. The ~たい form is used for actions that you want (or don't want) to do. ほしい is used for things you want to have and keep. However, if you were saying that you want someone to do (or not do) something you would use ~てほしい. 病院に行ってほしくない。I don't want you to go to the hospital.
When you add the ~たい ending to a word it becomes an い adjective. You add です to make it polite. The negative conjugation of い adjectives is to change the い to く and add ない (or ありません if you are being polite.)
ではない (and ではありません) is the negative conjugation used with な adjectives and nouns.