unter Betreuung der Bachelorarbeit schreiben
Die Arbeit is feminine and so die Bachelorarbeit is, too. This means der indicates the dative or genitive case. This marks die Bachelorarbeit either as the receiver of the möchte schreiben action, or the owner or an attribute of Betreuung. For die Bachelorarbeit, this doesn't make too much sense. Let's use another feminine noun, die Behörde instead:
Ich möchte unter Betreuung der Behörde schreiben.
I want to write to the office under supervision. (assuming dative is meant)
I want to write under supervision of the office. (assuming genitive is meant)
To express Bachelorarbeit being the thing you wanted to write, you had to use an accusative marker instead:
Ich möchte unter Betreuung die Bachelorarbeit schreiben.
Wahrscheinlich erinnern Sie ihn.
You cannot use erinnern this way in German. The accusative object is the person who remembers, not the thing remembered. It sometimes takes a genitive object but this means the same as the verb gedenken and is reserved for remembering killed people.
Using an accusative object with erinnern as the thing remembered is an anglicism and runs a chill in my spine because of both.
Wahrscheinlich erinnern Sie sich an ihn.
Sich is a reflexive pronoun. The person who remembers. The subject. The thing remembered is put as a prepositional object with an+accusative.
Sie haben mich mit ihm kennengelernt, …
You met me being with him.
Huh? WHAT? Well, because mit ihm indicates a relationship. I'm pretty sure that's not the case. You have to avoid this phrase then.
Sie haben mich dort kennengelernt, …
…, als wir nach Ilmenau während meines Praktikums in ihre Firma gegangen sind.
…, as we walked into your company during my internship after the Ilmenau incident.
Gehen means to walk in German and there are only a few exceptions when it means to go. The nach Ilmenau part has to be aligned to the other place in ihre Firma. If you put it in front of another time expression, you are talking about a time, not a place. But I think you talk about being at this company rather than going to it, so it should read
…, als wir während meines Praktikums in ihrer Firma in Ilmenau waren.