The directory sites/default/files does not exist I received the "The directory sites/default/files does not exist" error on a 7.43 install. In this situation, Drush is not available so I am doing a browser install.

I had this error once before in 2014 according to my notes. I don't know whether it's specific to the server I'm on or a general issue because I've done setups on other servers and don't remember seeing that particular error. In the previous situation, I noted that I changed the permissions on default to 777. I am always loathe to set much of anything to 777 so I am wondering whether it needs to be kept at 777 or is there some point after setting up the site that I can switch it to something more secure.
Your perms should (arguably) never be 777 on anything, but your exact situation will dictate that. 755 is often more appropriate for writeable folders, 644 for writeable files, but it depends on how your server is configured, and who owns the files.

You do need to create the sites/default/files folder manually, before the web-based installation can proceed, and your web server user must be able to write to that folder. You also need to create sites/default/settings.php, make it writeable, then remove those write perms once installation has completed. Drupal's status report will tell you about this if you haven't already done it.

There's not a lot more to it, just make sure the perms on all your files and folders are sensible for all the users that can interact with them. Securing file permissions and ownership has the full details, I'd recommended reading and fully understanding that page before going any further.