All the contents made with Toolset are stored in the Database.
All the contents will be there, if the exact same data of the exact same database is used.
It seems hence, during the migration process some of that data was either lost, or not migrated.
The data mostly sits in post, post meta and options tables, but also others.
Note that I do not see any search HTML of Toolset in the pages you linked. Only the loops are there.
Those are all Divi Builder created items, as I can see that in the browser console.
I suspect, there is a mixup in assigned templates but am not sure, see below.
1. In the related views, when you edit those, do the Search and other things still appear in the editors where they where coded into? I refer to the Search and pagination area in the View.
If the searches are still there there should be no reason for them not to display, unless some conditions are wrapping them, or others.
If the editors there are empty, the content is not in the database and is not recuperable on this very install. In that case, it would need the copy of the old, working site/database to re-migrate or re-instantiate it.
If you still have a working site access, you can export the views from there, and after deleting them on the new site, import them.
2. Adding links where previously there where none is not possibly done by Toolset, it is the first time I hear of this, and it is bot possible because Toolset does not create content automatically.
I suspect here something else could have happened, namely another layout is applied than before, or where thee was no layout, now there is one.
Are all the templates and layouts you assigned to the pages exactly the ones you expect?
If not, does changing the template to the one you expect fix this issue?
3. The markers issue will be solved when you loaded that page as many times as you have markers/10. This means, each time you load the page, Google will allow 10 more addresses to the cache.
That, if the Google Maps API is correctly configured and working, of course.
It is something that is often annoying when migrating, or clearing Maps cache, but cannot really be avoided, because it is how Google allows us to work with the addresses.
Can you try to load the page a few times to see if more and more markers become visible?
4. Now, at this point I am almost certain none of the issues are bugs in Toolset.
These issues are due to a migration gone wrong, likely, and it is difficult to determine why.
What I suggest is to either ask professional assistance to migrate the website, or you can follow the below instructions which allow you to use the Duplicator files you have, too re-instantiate the site.
1. Backup the current site
2. Save all your backups safely in a local location
3. Delete the current install (faulty site) completely, thru cPanel. Contact your host if you need assistance with this.
4. Upload the Duplicator Archive and installer.php file to your FTP where the install was (which you just deleted in step 3 above)
5. call yoursite.com/installer.php in the Browser and go thru the steps.
At this point you should have a site as it is in the Duplicator.
You can read more here about this as well
hidden link,
We cannot help with this in Toolset, but we can take the Duplicate you have, and test it locally, for example, to confirm wether it is working or not, and if not, wether there is a issue with Toolset or not (I suspect the duplicate will work fine, but in case it does not, we would of course investigate if Toolset needs to be adjusted)
Can you let me know if re-instantiating this site from the duplicator helps?
Please first make sure the content is not just accidentally hidden, or wrong templates assigned, as I above outline - it might save you the hassle of re-deploying the site.
Please let me know, I also activated a private reply for any duplicate in case.