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SEO Basics: Learn all the SEO basics back to front and well, don't just have a "general idea" of the basics. If you know all the basics "how long should a page title be?, What's a H1 heading?" you will connect the dots a lot better in SEO.

Reading: You have to (and should) read all the time and be on top of the latest news/strategies because it is always being refreshed and updated.

It appears that etsy provide information on this as well.

https://www.etsy.com/au/help/article/247
https://www.etsy.com/au/seller-handbook/article/4-ways-to-improve-your-seo-and-increase/22819094473

My interpretation of this is to treat your main shop page as a website and do what you would normally do to optimize a website.

I couldn't say why exactly but I believe that:

The photo might be from a website that's quite popular and Google thinks it's a better image? 
Have you uploaded the correct photos? for example, some people use their profile photo to place a photo of their logo instead of what's actually meant to be there (an image of what they're service is about).

I would go back and look at the example's in Google My Business Listing and make sure you've uploaded them correctly (or triple check) as well as your G+ page.

Create a list of questions that your audience would be searching for like EGOL said.
I do that by going into Google search and starting typing things like

What does type topic here
How will type topic here
How can type topic here
And other 'who, what, when, where, why, how, is there, can you' type questions.
By adding an underscore it usually finishes off the question for you as well as give's you a few more questions or phrases you might not have thought of. This is pretty time consuming though.
After that I put a list of the phrases in Keyword Planner or download the plugin Keywords Everywhere (Free) which will give you an idea on search volume which you can sort in order of importance.

Cheers,

Phase it to ensure every aspect of this migration is accounted for meticulously. A new domain move is always risky and then adding an additional redirect to the mix will make it messy and hard to keep track of.

You could potentially do the below:

Organise the redirect list and make sure every url is accounted for from the domain you're redirecting (I'm sure this is already done)
Change hosting providers first so you can manage and work on everything under the one account.
Add a 301 redirect script for the new domain to automatically redirect http to https 
Then redirect the old domain urls to the new https domain to avoid the additional redirect to the http

This would avoid the typical chain "http://olddomain.com > http://newdomain.com > https://newdomain.com" and instead just have "http://olddomain.com > https://newdomain.com"

Up to you how the process is done, but I would definitely phase migration.

I normally do a fetch and render in Google Search Console (webmaster tools) and submit to index. It gives you the option of submitting an individual page but I normally fetch and render the main domain and then select 'crawl this url and its direct links'. Then if there are only a couple of important pages not being indexed or recognized by Google, fetch and render and submit those individual ones.

type in site:www.yourdomain.com" in Google to monitor manually which pages are indexed as well.

If you have the time I would pop a list of your original urls in screaming frog to see which page their redirecting to, or test a couple of pages using this http://www.wheregoes.com/retracer.php to see how many redirects are in place.