We took a sampling of the pages on our site, 7.4 platform, and have noticed that each page starts with <H1> tag but the next category on the page is <H5>… we understand that Google rarely searches beyond <H3>.

If this is the case, then this might be a partial explanation as to why the actual number of page in the Google index for us is 486 vs. Google reporting of 793. We update content and submit site maps regularly and are perplexed why a fraction of our pages are in the index after 6 months. We’ve added < 40 new pages in the past 2 months, so this doesn't account for the differential.

Is it possible to change the secondary heading tags so they are <H2> or <H3> rather than <H5>?

Just out of curiosity, where did you get that information from?

I think the reasoning behind using H5 tags rather than H3's is it really makes no difference. What I mean is H1 will be your most important heading on the page, H2 sub-headings, H3-H6 everything else.

My guess is that Netsol made them H5 because if they changed it to a H2 it could be considered spammy if the person has a large number of category pages. Also it allows us to use H2-H4 tags in out long descriptions for the categories. If you wanted you might be able to replace the tag with javascript but I am not 100% sure.