For Linkearning you need content wich is readable/shareable/linkable - really great stuff with a lot of value for others, without great content it is a Mission Impossible. So if thats part of your "onpage" - you need great value to share with the right people.

It's hard to answer this question, we don't really know how far u are and what you exactly mean with onpage or what the site's content is.
I think Links to products are pretty hard to get. You need content round about your products, linking to them. Outreach for Links to products could make you sad and depressive ...

PS: I think Rands last WBF helps a lot in saving time by do outreach in a better way: https://moz.com/blog/link-building-outreach-in-a-skeptical-world-whiteboard-friday

Sounds like you have an mobile optimized Website (responsive, correct redirects, somthing like that) or your website has a pretty good User-Experience when it is used on mobile and some of your competitors don't.

For more Tipps we need more details

just to be sure - that all the domains "seem to be de-indexed" includes the one in question?

32 Redirects shouldn't be a problem. The domains with 301 couldn't be indexed - they are pointing to another location. Thats the same for 302, the question is the duplicate content. I have seen a lot of sites with duplicate content (means here really everything, content, style, etc.) and google just picked up one and indexed that one - not one other of the duplicates. One time it starts to de-index the indexed one and indexed another domain (thats another story) - I never saw 2 pages with the same content and style indexed at the same time in that case. The only problem should be, if you have duplicate content on a completly different site - different pages, different style, a.s.o. -> a "competitor" made by yourselfe. I never saw google de-indexing a site for that, but goggle ranked them lower and lower and lower.

If all pages are de-indexed I would guess there are more problems. If the one domain in question is still indexed, I would say thats how it should be.

When I compare image and web, I can see that image searchers aren't counted as web searchers.
When I check positions, I can see that some search querries of one of my sites are at positions 1.1 to 1.9 based on web and 3.1 to 45.4 based on image search. Thats working - as long as it is picture search vs. web search. And as long as the search querry has no image-box.

Now your question - I checked another side wich has a lot of rankings in image boxes in web search. And I think you are right:

• Search Console is telling me I am on the web at postion 1.1
• When I Google it, my picture is the first one in a picture box at the top of the Results.
• Than there is a website (wich is not mine)
• And finally I am on Positon 2 (Picture Box, other Website, my Website)
• the same with a lot of querries. Google is here counting my image result (first result) in a picture box @ web search.

Tried to find some clients - found one. A search result he simply couldn't have with his website. But he had - it was a picture somewhere in the middle of the SERPs. I also saw that google seems to count each image as one result.

So finally you are right. If you have a Picture somewhere in an Image Box (wich is displayed in a Web Search) is a Web Search Position in Google Search Console.

If you have redirected the URLs correctly you should fatch the homepage as Google in WMT and send them including all refereing links. the new URLs are pretty fast in SERPs than.

The rest is like Ryan Purkey said - thumb up

I think you could have is the following problem:

You got a Domain wich earned Links bevore and these Links are still out there. Google can see them and you will again and again see them in WMT. 
You cant just say solved and than they are gone, caused by the backlinks. They come again. 
Check that in WMT - I dont know how it is called in english versions (I just see the german), you can click on the 404 and than take a look at "what is linking to that page"
An easy solution in that case may be to disallow "/fs" for bots. (if you dont use fs in your url structure)

I cant say any other thing than @evolvingSEO he is right, your content clearly looks like social media. 
There is an opportunity for image search of course, thatswhy you could name the puppy/dog you write about like I said above - but your traffic screams out social media

If you optimize for dog contest, just because it is searched a lot, you will frustrate any person wich want to participate on a dog contest (as long as you dont do dog contests).

Yoast tells you what people search for, i guess noone searches for "Dog of the week". 
If you write for a targeted Keyword, you should do that only one time - otherwise, what should google show in SERPs??

But in your case, Dog of the week is not the targeted keyword. Don't know how people searching for dogs you write about, but that should be the target keyword - maybe for the category - and with the post about the dog of the week should target other things. Maybe "brown labrador retriever puppy"

2 things nobody said:

1st
Robots.txt says disallow to image and Theme (where i guess css and js is included)

2nd
WebOfTrust warning to not enter the site because of not trustworthy