It will take either one depending on your preference. 416 and 516 mean thickness of 1/4 (4/16) in and 5/16 in respectively. The 516 will be more responsive, though I’ve never tried it. The woodthread size is the same between the two. TMBR yo-yos come stock with 416 walnut.
The eH came with YYE cotton 10, which is the best string for fixed axle in my opinion, but they do not last like poly on unresponsive yos, you got to change them often.
I know my eH’s axle was catching on the string simply doing a trapeze for a while. I probably should have done the ol’ “pull an old string through the axle a bunch,” but I’m lazy and just threw it hard into trapeze until it stopped returning. Seems to have worked well.
Edit: YYE cotton 10 is dope. I have way less problems with strings snapping since I picked it up.