Garage floor drain and connections are dependent on local code. Around here it is common to do daylight drains. They are ok for residential up to specific size garage. If you are over the size spec, things change a lot. Separator setups are spendy. Going that route also puts you into a whole new set of code requirements. Bla, bla, bla. It's a bunch of details we don't need for the discussion.
First thing, look up local code. They probably default to another building standards code unless you're in a fancy city with fancy rule makers.
Looks like you are going to be around 1,000 sq ft with space. That may be a factor. Look up the code.
The questions about pipe size will probably not be covered in code. I can give some personal perspective.
You mention three inch drain piping. What kind of drain are you planning? The diagram shows a trap drain. I do not like trap assemblies. Around here they need to be in a conditioned space for freeze protection. They plug way too easy no matter where you live. I'd verify if that is needed.
Sounds like you are using smooth wall perf for at least part of the line. Definitely better than corrugated anywhere on the line. I would plan on non perf past the footing. Draining to a yard basin might be tricky. Over time sediment adds up. That option also goes back to the first point. In fact, most of the discussion goes back to what is required before we can talk about real options.