Version 5.2.0 was uploaded in March 2018 to replace 5.1.1 which had been available since November 2015. The only changes are to optionally support shortcodes in the text of widgets, and some fluffy bits that make the code more 'correct'. 5.2.0 dates from June 2016 itself.
Version 4.1.0 introduced the Panes module. Two were displayed on the Main Index page by default and should be turned off in the options if not required. This increased to three in version 4.1.1. Version 5 added an extra row of menu boxes which you should again disable if not required.
Version 5.0 changed the name of the theme directory to 'scoutingchallenge'. If you use a child theme you will need to edit the stylesheet in your child theme to pick up the new parent. You may also find widgets have been disabled or moved into the wrong sidebars. You may also find WP Menus need reassigning to the correct location. Apologies for any inconvenience this causes.
The download is in the format of a zip file. To install the theme go to your WordPress admin menu. From the menu choose Appearance and then Themes. You will see a screen showing your installed themes and at the top there is an Add New button. Click on that and on the next screen you will see an Upload Theme button at the top. Click on that and you will see the instructions for uploading the zip file you have downloaded. Once the theme is uploaded you will be able to activate it.
Version 4.0.0 should be compatible with the 2015 Brand Guidelines (and if it isn't then that is something that can be looked at). The previous guidelines were published as one 'corporate' set (for want of a better description) that defined primary and secondary colours, plus separate section guidelines with other colour sets. The latter had not been incorporated into the theme. There is now a single set of guidelines and one colour palette, but sadly no names for the colours, and although it was possible to map the old colours into the new palette there will be obvious differences. Purple, green and grey are the same (though only the one grey is supported now). Orange and brown haven't changed much but you might find blue and mauve have changed enough for you to want to use something different. Perhaps surprisingly black has changed too and is now defined in HTML as #000000 which you might perhaps have assumed it always was. You may find what the theme calls turquoise, green, teal and dark blue are a useful match for the Beavers, Cubs, Scouts and Explorers logos, and dark green is presumably the Cubs uniform colour.