Each time I try and run BlackBerry Desktop Manager I get an error message and a reference to a Microsoft website that says, "Blackberry Desktop Software for v3.6 and earlier has stopped working properly", and provides a link to download BlackBerry_Desktop_Software_v3.6_SP3a_ML.exe. Is this the latest version of BlackBerry Desktop Manager and do I just run the exe file to replace the current version I have which is 4.3.0.15? It seems like Microsoft is suggesting an earlier version.

no, the latest version is DM 5.0, download and install it.

BlackBerry Desktop Manager Downloads - BlackBerryFAQ

It says I should check with my carrier (Verizon) to see if this version is compatible with my device. Is this necessary?

It is compatible. I wouldn't advise you to do were it not.

Nope.
Go ahead and download ver. 5

It should be compatible for syncing with Outlook as long as you have Outlook 2003/2007. If you use Outlook 2000 or earlier, then you need DM 4.2 SP2/SP3.