All your questions predicate/depend on site's monetization strategy.

For example, if current pages all have great SEO traction in SERPs than any site change best be well thought out + in general the HTML structure should...

1) Only fix any existing HTML errors, as reported by the W3C validator.

Never. Ever. Introduce any new HTML errors or site's SEO can circle the drain.

2) Only lower page weight, so the ratio of cruft (non-content) to content, should reduce. So decreased cruft, which will increase content.

For example, converting a well SEO'ed WordPress site to use a Genesis theme can potentially destroy all your site's SEO traction, because Genesis (at least last time I checked) can produce a site which is 98% cruft + 2% content, due to all the CSS classing + attributes + other non-content junk injected into pages.

When I take on a project for one of my clients like this, first I require them to host with me...

Because wrestling with broken hosting introduces so much noise into analysis, I just refuse to deal with slow + glitchy hosting anymore.

Next I have my client walk through their entire money flow, end to end.

After I understand site monetization strategy, then we work through small increments of change, constantly tracking when GoogleBot visits the site + how page indexing is effected in searches after a 48 hour period.

If it's a hobby site, all this is overkill.

If the site measures profits by the hour or minute, then any change can potentially zero out all income, so in this case best to proceed slowly, with great care.