1) In your experience, how does branding impact SEO/organic traffic?

There are a few ways to look at this:

[Established Brand = Increased Traffic via Branded Search & Navigational Queries]
Looking at the analytics of a site with no brand vs a site with an established brand you will see a significant amount of navigational & branded search queries. This is great for business however I typically filter known branded terms and traffic to the homepage + landing pages known for serving branded queries as direct traffic so that I can better focus on non-branded queries.

[Trusted Brand = Higher CTR]
The real advantage of a trusted brand comes when someone does a search for a non-branded query and chooses your article in position 3 vs a brand they do not recognize in position 1. This in turn sends positive engagement signals back to Google which can help with your positioning.

[Authoritative Brand = Higher Return Visits from Search & Browser-Based Bias]
If someone bookmarks your site or has visited a page on your site recently, when they start their search query (often from the URL bar) it can pop up with past content (although this would show up as Direct traffic). Also, there is a correlation between branded search queries with positive engagement signals and impressions for non-branded queries.

2) What are some ways to leverage branding within SEO strategy?

[Structured Data = Branding in SERP]
Leverage structured data and 3rd party validation (over the years it has been Wikipedia/Freebase/DMOZ) to add your “Site Name” after your URL and other enhancements. Utilize schema rating to stand out on product based queries. Call further attention and provide greater context to your listing with breadcrumbs, sitelinks and searchbox.

[Google Search Quality Guidelines = Expertise + Authority + Trust]
Brand building focused on authority is a pilar of Google’s Search Quality Guidelines.

[Meta Description = Sales Copy]
Leverage the meta description to write compelling sales copy for each landing page, don’t forget positioning/benefits statements and a call to action!

3) What is the best way to measure the success of your brand online? Feel free to mention specific KPIs or name drop tools.

[Chained KPIs]
There are a number of factors to measuring your brand online, specific to SEO, it would start with Branded Search Demand (Google AdWords/Trends/WordTracker/Moz/Bing Ads Intelligence), then SERP Performance (Branded Impressions/CTR via Google Search Console), check content performance through to conversion from Organic referrals, check Inbound Links (analyze both branded and non-branded hyperlinks via Moz/AHREFs/etc.) and referral Traffic (Google Analytics). Outside of organic/SEO, look at branded social mentions & engagement (Native/HootSuite/SproutSocial/Mention, etc),

[Google Search Console = Analytics for the SERP]
As mentioned above, Google gives us analytics tools to see how our brand is performing, place more emphasis on KPIs you can directly control (CTR) vs. KPI’s you cannot control directly (Ranking).