Hi there!

I hear you, had some simililar issues with big sites that i've worked with.

I would go with the "highest page viewed" page to be the canonical, you should select the one that you consider serves users with the best experience or match their search intent. That said, i'll answer your questions:

1. No, there is way to get that many URLs from Search Console. Through API the most I could ever get was arround 200k URLs.
2. It's not that hard as "overriding" the actual canonical page selected by google... You should tell Google wich is your preferred canonical page, then G will decide whether to use it or not.
3. Is always wise to go first with a canonical, so you can analyze how google reacts with that change. Then wait a little time, a few weeks and if everything went as planned set redirections.

Hope it helps.
Best luck.
GR.

Hi Matt,

You're welcome, we are here to help.

Yes! When a migration is done, always redirection 301. Remember to map all the "old" URLs and the "new" URLs must match.
PLEASE! Do not redirect all to one page.

Best luck.
GR

Hi Matt!

From se SEO perspective, you could do either Redirect 301 or a Canonical tag.

On one hand, making a redirection could make User experience worse as you are forcing them to a page that they might no be expecting or looking for.
On the other hand, as this are parameters that you don't want to be in Google nor being SEO friendly, then using a canonical tag is the answer, this way you tell Google that you have a better page for answering that user intent.
Here some resources that might help:
SEO Best Practices for Canonical URLs + the Rel=Canonical Tag - Whiteboard Friday Consolidate duplicate URLs - Google Search Console Help

Hope it helps.
Best luck!
GR

Hi Risto,

Google said it many times that they are pretty good at detecting bad links thus they encourage us to disavow links only when a manual penalty has hit your site or when you dont trust Google's algorithms and/or do know that those links are really hurting you.

That said, my personal opinion, don't bother at all. 
Also, please keep in mind that private metrics as PA, DA, TF, CF or whatever do not reflect what google sees. Those companies with their algorithms try to simulate what Google understands so don't fully rely on them.

Hope this helps.
Best luck.
GR

Hi angela,

Those 5 out of 64 URLs.. Is that a report in Search Console? or only 5 URLs appear when searching in Google?
Search Console usually takes a little longer to update its reports on index status.

Have you tried a site: search? Also using _inurl: _parameter.
For example: site:domain.com inurl:/category-noindexed/

Hope it helps.
Best luck.
GR

What sometimes happens is that when some URLs are marked as noindex, googlebot reduces its crawling frequency as they interpret that you really don't want that page to be indexed thus has value or so ever.

What you just did is tell GoogleBot to crawl specifically that page and "force" it to analyze and render that page. So GoogleBot now understood that the noindex is no longer set and that page should be indexed.
I'd wait a few days so that googlebot naturally crawls all your site again and eventually index every page that deserves to be indexed.

If that doesnt happen in about 2 weeks, then there is a tool in the old Search Console, where you can tell GoogleBot to Crawl a single page and its links. That is under Crawl-> Fetch as Google. Request an URL to fetched, after a few minutes it a button: _Request indexing_will appear, there you'll have the option to "Crawl this URL and its direct links". This image might came handy: https://imgur.com/a/y5DbUVw

I'm glad it helped previously and hope the last helps you even more.
Best luck.
GR

You're welcome. 
We as a community are here to help.

If your issue is now fixed, you could mark this question as answered

Best luck.
GR

Hi there,

I don't fully understand what's your issue with canonicals as you made it look like the issue is related to hreflang tag.

So, answering your questions:

1. Canonical tags should point to a page that means the same INSIDE the same website. If that's not the case, its a common practice to have self-referential canonical tags.
2. Not having canonical tags at all **will not impact **your rankings nor its ability to rank higher. With this, you are telling Google that you don't know if there is a better page and let google choose for itself what's the best canonical (if any)
3. Having different domains for different countries and even with different letters/words is not an issue at all. You have an advantage having ccTLD domains for each country, this means that Google already understands that your website is targeted to that country. If between both websites, you share content and have a set of pages that mean the same but for different countries, then you should add hreflangs so G gets the idea that both sites relate.

Do these comments answer your questions?

Hope it helps,
Best luck.
Gaston