Blu-ray

"If I output to a MP4 and burn to BD, will that preserve all the menus I've created in my video? I'm thinking it won't"

You are correct.  Menus are not part of the MP4 structure.

"I was trying to make a AHCHD disk on a DVD which Adobe says you can do..."

Adobe used to say that up to about two or three versions ago.  It worked very well on a Blu-Ray burning laptop I had.  Now Adobe says, "no Blu-Ray support in any Adobe program".    On Windows you can still make DVDs.  On Macs you can't even do that.    

It appears that Adobe believes discs are going away due the popularity of streaming, Netflix, YouTube, Vimeo, 4K, memory stick TVs, etc.  

I gave up Blu-Ray when my burner quit and I decided on a 4K work flow.  So, I have no recommendations.  That said, I read that "ImgBurn" is free and works.  DVD Architect is not free but originally developed by Sony when Sony was pushing Blu-Ray hard.  It may be the best!

Google search for "does imgburn burn blu ray".  I got a lot of positve hits. 

"I spent a lot of money on all of that."

If Blu-Ray is your primary goal, Premiere Elements is the wrong editing program.  Adobe made a corporate decision to not support Blu-Ray burning in any of their software.  It is history being replaced by other systems of distribution.

If you bought directly from Adobe, you can get a refund if you act soon enough.  Contact Customer Care.

https://helpx.adobe.com/contact.html

No. That's no longer a feature of Premiere Elements.

However you can output a high-def MP4 from Premiere Elements 2019 and then use a third-party program like DVD Architect to create a BluRay from it.