No you cannot have one layer using maplex and another layer using standard labeling within the same data frame.
However, you could enable standard labeling in a separate dataframe within the same MXD. That way one dataframe would be maplex, and the other would be standard
I don't see why you would want this though, so perhaps if you provide more additional information to your post I could recommend other solutions and give a more thorough answer.
I think labeling comes down to your own personal choice. However, I personally, prefer maplex for labeling. I noticed in your screenshot you just have "regular placement" for address parcels usually I change it to "land parcel placement". You can also click on
properties.. to get more options for placement, even resizing text to fit, specifying how far a label can extend past the parcel boundaries, and what labels or shapes hold more weight (so they won't overlap or extend over onto those polygons/lines/points/labels/shapes.
This does a better job outlining Working with Maplex and the Exercise Labeling the parcels with their serial and parcel numbers
As per your additional update, one thing I noticed under "Fitting Strategy" is you don't have "Reduce font size" selected. I would check that off and specify your lower limit (how small text can be) and as well, change the strategy order so that reduce label size is on top - and play around with the order of the remaining. You could also change the "Feature Weight" of the roads adjacent to your parcels, so that your labels won't overrun on top of the road (and therefore be forced to fit within the polygon boundaries).
Another, additional option if you're only having issues with certain polygons is to create additional label classes and use a SQL query to specify them. Further to this you could always convert to annotation.