I'm not a Mac user so probably not the best person to help, but I'll give it a shot.
1. first question - I can access photos from Photos, Aperture, and Lightroom. I can't clearly figure out where these files are kept. If I actually have 3 (or even 2) copies of each photo, I would like to eliminate the extras and just rely on Lightroom.
The actual structure depends on the options you chose during the conversion. Scroll down to 'Set Your Options' here: http://photofocus.com/2014/12/15/the-essential-guide-to-moving-an-aperture-library-to-adob e-lightroom/
It should be fairly simple to determine if it created duplicate copies and the ones you can delete once you understand the options you chose.
2. I decided I could see some folders that are being access in Lightroom. They are located in Mac file Pictures in file structures that match that in light room (i.e. Family with subfolders structured as they are in LR). I think I made a mistake adding pictures to those folders assuming they would show up in LR - they don't. Now I am thinking that they should be imported. Can I just import a folder and it will just pick on the ones that are have not be already imported? It would be very hard to identify by looking which ones are already in and which aren't.
I assume these are new camera files that have not been edited yet. The simplest way to do this set the option for 'Show Photos in Subfolders' in the Library menu. Next select your top-most level pictures folder (named Pictures?) inside the Library module and then right-click and select 'Synchronize Folder.' . The "new" image files will be automatically imported into LR and appear in their respective folder locations.
3. It seems that no matter how I structure folders there is also (within the folder or without) a separate file organized by date. I assume that is not a duplicate but it is confusing. Is there a way to just look at the file structure I created and eliminate the date files?
If you have 'Show Photos in Subfolders' checked in the Library menu it shows the photos in the selected folder AND all of its subfolders! Try unchecking that option in the Library menu and see if it resolves the conflict. Also scroll down to 'What About File Management here: http://www.lightroomqueen.com/ready-move-aperture-iphoto/ Again, the creation of "duplicates in dated folders" is an option that can be selected during the Aperture conversion.
4. Is there any easier way to interact with LR on Mac? It seems I have to export a photo to the desktop then drag it into an application.
LR is a non-destructive editor just like Aperture. The original files are never "modified" or written to with the LR edits. You must use the Export module or 'Edit In' function to create an edited "copy" of the original file for use in another application. Lot's of good Help and tutorials here: Learn Lightroom, get help and support | Adobe Photoshop Lightroom 5