If you are going somewhere like a hospital often, i would think that you are literally going inside, and with another person... I get the sense from it like, you or she is ill or visiting someone there. That's why, in this context, i would use "ins". Additional to the translation:
Aus diesem Grund muss ich oft mit ihr ins Krankenhaus (gehen).
First, keep in mind that there's no direct correspondence between prepositions; you have to look at the exact meaning and figure out which preposition is the best match. (This is true of most words in fact, but prepositions are often particularly tricky in this respect.)
German allows you to drop the main verb when there is a modal verb (such as müssen) and ...
Talking about a named company like VW or Aldi, you will use the preposition bei:
Ich bin Praktikantin bei VW. Ich arbeite bei Aldi als Kassiererin. Er ist Techniker bei der Telekom.
If the listener(s) are working in the same company as you, you can say
Ich bin hier Praktikantin / Ich arbeite hier als Praktikantin / Ich in hier als Praktikantin eingesetzt.
If you want to translate your given sentence word by word, it would be something like this:
Aus diesem Grund, muss ich mit ihr das Krankenhaus oft besuchen.
If you visit something, you don't go into (ins) or to (zu/zum) something. You just visit it. For example visiting a graveyard, or a friend.
But if you go to the hospital, you would say Ich gehe zum ...
You are right, the translator is wrong. Your explanations are correct:
[...] [W]hile using company, we should always use bei.
Note: von would sometimes be an alternative.
female intern = eine Praktikantin
I would prefer
Soll ich Dr. Mueller schreiben, um an meinen Brief zu erinnern?
It is also possible to use
Soll ich an Dr. Mueller schreiben, um an meinen Brief zu erinnern?
You can use jemandem schreiben (Dativ) or an jemanden schreiben (an + Akkusativ).
It must be "…, um an meinen Brief zu erinnern".
Especially in this example sentence the ...
The question why there are spellings like Chemnitz, Chiemsee, Cham requires a historical explanation. In the course of the High German consonant shift, Germanic P T K turned into F S CH or PF Z KCH in High German. Modern standard German is based on varieties that have not completed this process. Initial K remains unshifted in modern standard German. Only the ...