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"The site had amazing rankings and traffic previously but is really struggling to regain rankings now. "

And what happend to the backlinks? Taking over an expired website is always tricky.

It's crawler is relatively slow, and because of that old data might appear still. It's normal.

Did your rankings actually got better? Because DA/PA is completely apart from Google itself. Your not hiring a SEO company if the goal is to just increase the DA/PA.

If it's indexed and in search results the brief answer is yes. But keep in mind that google webmasters is sometimes behind to actual data happening in search.

You could do "site:yoururl.com" in google search to see what's indexed in Google.

Update frequency, i.e how many times your updating, and value of the website for google.  Ive seen my own website being crawled daily at some point.

It just indexes your pages. That could be done in less then a minute if it wanted. If you have everything structured with for example an up to date sitemap, no 404's or anything then your good to go really. Crawl speed is a factor on how much time it spends on the website. Crawl speed is mandatory when google is requesting alot of pages at the same time that could slow it down, or be triggered by a firewall for having too much connections at the same time.

Really these things are usually something from the past. If you want a quick index throw in a link on social media for example or get a quality link from some other place that's indexed more often.

I have the same going on with one particular keyword. The best way is to get a good link for that keyword thats having a lower or different ranking then the other keyword. Then it should be solved.

Are we taking these spammy posts seriously?

If you update your website 'frequently' the crawler will be more there, if you dont update your website frequently the crawler will slowly back down. Ive had a client's website not updated in perhaps 2 years. We installed a complete new website with new content and it took months for it to be completely re-indexed.

You could assign a priority to the page by using your sitemap, but on the other hand the priority is sometimes completely up to google.

Yep, add to it, labels are not the #1 argument a website is being listed or shown in search in the first place. So dont put too much of effort into a label really.

Take into account that, both MOZ and A hrefs have the tendency to still list links while being removed for a while.

Really these numbers dont mean alot in comparison to google. Both are not asociated with google but do their own maths or calculation on how to summerize it's DA/PA. What google does is litterally a mystery for everyone except for who works at google obviously.

https://moz.com/learn/seo/canonicalization

Yes, that "loading" score is horrible. Just 8 seconds to load the first few batches of threadwork (javascript). Imagine the experience for a mobile user on limited data or so.  Offload as much as possible of scripts; really having a neat website does'nt require so much bells and rings all over the place.

It's a myth that your DA drops because you put links in disavow. Disavow is a google only (or bing) tool, where lets say you get spammy links from a rogue domain and there's no way you can get 'm removed.

MOZ cant read your disavow file either you file into google. So i'm not sure on how the link is being put here. With MOZ, or any other tool, they just calculate the amount of incoming, FOLLOW links and presume your DA on some magical number. Thats all there is to it. Again, PA/DA has nothing in common at all with Google as Google maintains their own algorithm.

Links really need months in time to be really effective. So as long as your losing links are not that much of a value, your good.

Dont overblow it really. I'm working on that too right now with positive effects, i.e /subject/another-subject/, it would be good if you link all the independent pages from /subject/ as well including a dropdown menu on /subject/ with all /another-subjects/.

https://moz.com/domain-analysis

Type the domain you want; underneath the output is the PA per page, basicly listed as highest PA per page to lowest.