I am a non-EU student currently studying in France, where I am residing with a French residence permit (titre de sejour), which in theory lets me do a 90-day internship in Germany. I have received an offer to do a 6-month internship in Germany, but the start date is too close and I am worried my visa will not be processed in time. Would it be possible to start the internship and work for up to 90 days until I receive my visa, and then finish it when/if I receive my visa within those 90 days? If not, is it possible to sign a short-term internship contract (60 to 90 days) and then sign another, longer-term contract that starts as soon as the first one ends, and do the first one without a visa while I wait for the visa for the second internship to be processed? Thank you.
No. The Schengen (tourist) visa does not allow you to work, and cannot be converted to a work visa. This is explicitly disallowed, and some immigration offices (like Berlin) explicitly warn you about this when you apply for a residence permit.
§5 Abs. 2 AufenthG: https://www.gesetze-im-internet.de/aufenthg_2004/__5.html