I always recommend that the h1 tag be an expansion of the title. The title is limited by the number of characters (45-55 or as long as 65) to be effective as a SERP link while the h1 does not have the same limitation. For this reason, when I write a title tag, I keep it pithy and word it in such a way that the h1 would be a longer version of the title tag.
As for branding your blog, you can brand in the title using either the
| character in the title tag, however, what works best is schema mark-up. I would suggest using both if you can. I do not use
| in my title tags at all. The reason is simple. I need the space for the title. I do use mark-up and branding works fine for me. But then again, my sites are businesses and not a blog per se' though there are blog style posts on the site.
Here is some of my answers explaining
Does changing company/brand name in title have SEO issues?
Site URL being displayed before TITLE on SERP
This may be confusing, however, in reality it is really simple.
Basically, you would create your title as something like this. "How to change a faucet | example.com" This would create a SERP link as "Example: How to change a faucet." You can also use "How to change a faucet - example.com" which will taken as is to produce a SERP link as "How to change a faucet - example.com" and still take up example.com as branding. I suggest using the first example.
As far as branding the blog, you can use the Schema Blog mark-up found here https://schema.org/Blog.
If you are an organization, and I mention this just in case it applies, you can use Schema Organization found here https://schema.org/Organization. There are other types of organizations listed toward the bottom of the page.