Apologies you have had these problems ... To answer your points ..
GO Button and PDF Buttons
Yes the GO button is an alternative to the Html button... and should just produce HTML... i admit this is confusing and i am intending to remove it. With the version you are on im confused why pressing html produces a pdf report, i've not seen that before. With the pdf button producing a blank Im suspecting your godaddy environment is not allowing the pdf converter to work.. can you do me a favour and edit your start.php script and find the line that looks like ..
// Required PDF Engine set -- to phantomjs or tcpdf
$reportico->pdf_engine = "tcpdf";
if it is not set to tcpdf can you do so and try again.. if that fails do you have the ability to go into the browser inspect/debug (right click inspect in chrome), choose network tab, run the pdf, click on the request that appears and show the output. That might give the error.. (sorry this is so complex )
Group Header issue and documentation
I would say the documentation is inconsistent and and that you have installed it correctly but the group header/trailer issue is a bug. Out of interest which path did you hard code to tmp? Also is this a feature you need or is it making you doubt the installation becuase of the other issues?
To be honest I am putting all my efforts into a version 7 to fix all the inconsistencies in the version 6. Version 6 was an attempt to make reportico compatible with Composer and make it work it with later php versions but I decided to now convert Reportico into more of a reporting framework and designer with a better documentation set. I have completed the work on this and am in the process of packaging it up for release.....
I realise you have had issues with the current version but i would be very keen to give you a link to the reportico7 to try which should improve things significantly. If you have existing reports you would be able to just copy them over into the new installation. I will get a link available hopefully this week and if you want to try it i would be be very happy to receive feedback.