Text Logo Process

First I would not recommend PNG as a format for print. You should be making print logo different than weblogo. Please consider if you should be using SVG for logo. There is also the Asset Export panel in Illustrator. (PDF would then be a reasonable variant for print)

If you are planning to use photoshop as part of your workflow I would just place Illustrator object directly in Photoshop as smart object. PNG has only Alpha transparency and no blending modes. (Photoshop is not a good option for making logos for Print)

If you want pixel perfect you can align to pixel grid. If you should expand depends on the logo, aligning to pixel grid may distort the graphic, since it is rounding to closest pixel.... so pixel perfect art must be designed to be pixel perfect from start. 

If you're working with text and want the end result to look correct in print create as much of it as possible in Illustrator. Most logotypes, brands, etc are created as purely vector-based items. Any live type is converted to outlines to avoid font substitution issues.

Beware of baking Photoshop bevel and emboss effects into something like a text logo. Those are pixel-based effects and they don't necessarily scale so well, especially at small sizes. Even if the effects are visible/legible too much of that stuff can plunge a design into cheesy territory quick. For print-based work it's more cumbersome dealing with pixel-based effects imagery that has to be clipped into the vector paths of a logo or other graphics elements than it is dealing with vector-only elements.

The 600ppi thing doesn't mean much (other than bloating the file size). Your "master" design should be vector-based and stored in a CMYK document. The raster effects setting probably doesn't need to be more than 300ppi. When designing for web, it's really best to design for specific pixel layouts, like 600px X 200px for example, rather than winging it with a ppi/dpi setting. Many people say screen based designs should be 72ppi because that's how it measures out on a display. Many computers displays have varying resolutions. My 17" notebook has a 3840x2160 layout; it packs considerably more than 72 pixels into an inch (it's closer to 256ppi). The ultimate thing that matters is the physical sizes in pixels of the PNG images, not ppi ratings.

To get lettering and other elements to look their sharpest when exported the lettering and other elements need to be sized and positioned specifically for the pixel grid. You can set capital letter height sizes in pixels within Illustrator and use snapping functions to get elements positioned just right so fewer edges are fuzzy when the PNG image is exported.

Also, you might consider using SVG as an option. The vector format is well supported by leading web browsers now. We don't have to worry about the tyranny of Internet Explorer anymore. SVG conforms more easily to high ppi displays and liquid web layouts.

And since I am only using PS for the top text do I then import top text png from PS, open in AI and group with other bottom text and expand and export same - anti-aliased, 600ppi?