I've drawn Rodger before during Halloween along with Eclipse and him sitting in an Ichor puddle, you can check it out too!
As for my thoughts and headcanons! I think he would have a sharp eyesight, because he's a magnifying glass which literally makes you see better when you use it. Why does he use another magnifying glass when he investigates? Well, he is a detective after all! I think it's pretty much tied to his entire character.
His handwriting isn't great and sometimes he can't make out what he wrote. He sacrifices pretty looks for speed and efficiency! He has so many thoughts and clues to write down so there's absolutely no time to write it neatly! And making sense out of these scribbles is a Future Rodger problem anyway.
Rodger doesn't have a mouth nor a nose so it's safe to assume it's hard for him to breathe, it's similar to asthma. I actually think a lot of Toons had problems with breathing, just like Dandy did in official tape recording. But with Rodger Delilah couldn't really do much, it was pretty much a design flaw. That's why Rodger is slow, he just can't move faster without harming himself.
You'd expect someone as Rodger - a dedicated detective who will do anything for the research - to spend nights without sleep and to drink a lot of coffee. But surprisingly, he doesn't! His sleeping is healthy, he doesn't have nightmares and he wakes up feeling refreshed after each night!