Conversions were/are popular because they made the best use of what was around. Left-over parts, if you will. Sort of like taking the turkey from Thanksgiving and making turkey sandwiches for a few days, then turkey soup after that.

Starting a project from scratch to make a conversion is like going to the grocery store and shopping for leftovers when the main ingredients are just as cheap.

Unless you are really keen on using that bike as the basis of your project, I'd suggest starting with a proper fixed gear or track frame as the foundation of your project. The bottom bracket shell looks rusty, anyway.

There are enough cheap COMPLETE fixed gear bikes around that you won't need to start building from leftovers. $300 shipped and you can be rolling 1 hour after the UPS truck leaves your house. Using my turkey analogy, this would be a can of Campbell's Turkey Soup. Not amazing, but cheap, fast, and easy.