The symptoms you are observing don't have anything to do with LR.
There are two different sets of date/times associated with a photo: the file times maintained by the operating system (Mac OS) and the metadata times that are stored inside image files and are set by cameras and software applications like LR.
The Finder shows you the file times -- the time that the file was created and when it was last modified by an application. Different applications handle the creation and modification times differently, and not all backup/sync programs will maintain the creation and modification times. The creation time shown in the screen shot, 12/31/69 7:00 PM, is a special time -- it is 1/1/70 0:00 UTC (assuming your computer is currently set to the EST time zone), which is represented in many programs by the number 0. This suggests that one of the programs you used to copy your files has a bug and set the creation time to 0.
LR doesn't set file times -- when it creates a file or modifies it, Mac OS will set the creation or modification time to the current time automatically.
LR shows you the times that are stored inside the metadata section of image files, and the most important one is what LR calls "capture time". Cameras set the metadata capture time from their built-in clocks, and LR lets you change the capture time using the Metadata > Edit Capture Time command. LR displays capture time in the right-hand metadata panel and next to thumbnails in the Library.
If an image doesn't have a metadata capture time, then LR will use the file's modification time (older versions of LR used the creation time). Scanners typically don't set the capture time, so in your screen shot above, LR is displaying the modification time next to the thumbnails.
In general, it's dangerous to rely on the file times to represent information about your images -- too many programs will do unpredictable things with those times. Use programs like LR to set the metadata date/time of image capture (when the shutter was pressed in the film camera). Industry standards also define an additional metadata date/time, usually called "date/time digitized", which is when the original image was digitized, that is, converted to digital bits by a scanner. LR doesn't provide any commands for maintaining date/time digitized (it's primarily intended for images captured by digital cameras), so if you need to also maintain date/time digitized, you'll need to use another program such as Exiftool (free).
Thank you so much for your lengthy reply. It really helped.
Thanks for helping me let go of caring about date created / modified times. I agree, they're subject to too much change from all of the different apps they go through.
How would I go about setting the metadata date/time in LR though? I'd be happy to do this for the problem images.
You can set the metadata capture time (when the picture was taken) in LR by using the Metadata > Edit Capture Time command. This date/time will be stored in industry-standard format and recognized by most other programs and Web services.