The screenshot says it all: You are already using the maximum allowed daily API calls allowed by your subscription plan. If you still need more than that, you need to pay for an upgrade of your subscription.

There is a Form but they do not approve extra quota at the moment. Maybe in the future. Best would be to contact your Google Account Manager (If you have one assigned)

https://docs.google.com/spreadsheet/viewform?formkey=dDdnNVF3aGpuLXV1R2V2Nzg3QjJoZWc6MQ

After looking at numerous answers on SO on somewhat related questions, I think I have a general idea of the answer to both my questions.

1. "How is it possible that the limit for queries per minute is 1.6m and the queries per day limit is 10k?"

   It seems the answer here is simply that it's not expected to be possible that the queries per minute ever reach above 10k in a single minute, and therefore it's also not expected that the queries per minute reach the allotted 1.6 million.

   However, the secret sauce to this pie of confusion is the fact that you can't change the queries per minute (or queries per minute, per user) but you certainly can change the queries per day (by direct request to Google).

   What this means is both queries per minute and queries per minute per user are pretty much useless to you until you request a high enough queries per day limit for it to even matter. For now, you can imagine your queries per minute and queries per minute per user are equal to the queries per day (which is likely 10,000) until your queries per day exceeds that actual quota for the other two.

   As discovered in the comments of this answer, when the documentation states "10,000 units per day, an amount sufficient for the majority of our API users" they are explicitly referring to "in the development phase".

   Google expects all API users to request an increased queries per day quota after development is finished. Therefore, that "10k" will become a couple of million (assuming your request is approved) which would therefore make the queries per minute and queries per minute per user matter a lot more.

2. "How do I make use of queries per minute per user?"

   It seems the answer to this question is: "You already are". As in, you already are using queries per minute per user but Google just doesn't tell you... Originally, when I asked the question, and I assume others had the same thought that the - meant that no "user" had made a request. As in, somehow the project was making all the API calls on behalf of the user... rather than the user making the API calls themselves, and therefore it wouldn't count as per user.

   But that's not the case, it seems that Google does count it as per user but simply just hides the information from you.

   I only indirectly figured that one out thanks to Linda and the image the OP posted.

I'm not going to mark this as the answer since I'm not confident it's actually all correct and there's an inconsistency with the 2nd answer. This was confirmed by Linda's answer.