Sitelinks tend to take awhile - I think they may need to be "earned" either through increasing branding or increasing general domain authority. Not certain about that, but I don't think they show up for newer sites often. The could be a factor.
I would build up internal links as well as branded links to the site, and I imagine they'll come soon enough. That Google article I linked to gives a small amount of guidance - since you're doing a static site I'd double check these items:
"There are best practices you can follow, however, to improve the quality of your sitelinks. For example, for your site's internal links, make sure you use anchor text and alt
text that's informative, compact, and avoids repetition."
For the red arrows in that seroundtable post, that's common for branded searches. It looks junky because the titles and descriptions (IMO) are too similar.
All of that said, I wouldn't personally worry about it for branded searches. If they've made it that far, they'll find you.
"Keyword | Company Name" is fine for your title formatting. Whether the homepage is different depends on your niche and the value of your brand I guess. Overall I'd probably leave it as keyword first unless you've got a great brand that's been built up quite a bit over the years.