It depends on the phrase. Prepositions of course have no logic, sadly. And they are not always interchangeable. I think it is much more common to say either we detect cancer at an early stage or in the early stages. It is possible, but much less common I think to say that we detect cancer in an early stage or at the early stages. But we could also say: Company X secures $350k for solutions [or research?] to detect early-stage cancer. Although it is getting long, this would be my choice: Company X secures $350k to develop solutions for detecting early-stage cancer. A medical person might put it: Company X secures $350k to develop screening methods for early-stage cancer. So many headlines, so little time!

Both "in" and "at" can be used with "stage", according to oxford dictionary. 1. There is no need at this stage to give explicit details 2. I was in the early stages of pregnancy.

I use “in”. Not sure about the theory behind it however.