Heroku offers PostgreSQL only, so you might want to switch over
in development to avoid any incompatibilities.
That’s not entirely true, Heroku offers an addon through ClearDB for MySQL. Also you can use Heroku with an Amazon RDS instance too (obviously additional setup required)
Since scale is an issue, you might want to test postgres & mysql itself for the specific operations you are doing (at scale), then choose the database based on the results of that test.
How can I address this amount of inserts per second into the database?
This seems like a perfect example of premature optimization 
If you’re really concerned, set up a test app on Heroku and fire up
jmeter or ab or something and see exactly how it performs. You may
find you have nothing to worry about.
Yes, I agree, although if it actually is thousands of web requests per second it will require more than 1 dyno, or a larger Heroku dyno (like the Performance dynos), and it certainly will hit some bottlenecks at some point.
I wouldn’t agree that thinking about how an app is going to scale (and how to architect it so that it will scale) is necessarily premature optimization, although over-architecting for scale before load testing to determine where the bottlenecks are certainly could lead to premature optimization (particularly, optimizing things that don’t need to be optimized). I think the point is, make sure you can’t disprove the hypothesis, “If I optimize X, I will see performance gain Y” for any specific part of the stack you might optimize. If you can’t disprove (and of course you must actually try to disprove) that hypothesis, it is reasonable to conclude you have identified the bottlenecks and can justify spending time on optimizing that area of the stack.
Since no one else mentioned it, I would add you will use New Relic significantly here. You will need to get the New Relic “plus” or “premium” plan, so you can drill down into the requests to see where the bottlenecks are.
By “ab” do you mean “Apache Bench”?
I think both those tools will generate request from the programmer’s connection. One thing I like about blitz is that it can hit your website with thousands of concurrent requests from different data centers across the planet, so you can see how it performs different depending on geographic region.
-Jason