Upgrading Drupal

I'm tasked with updating a Drupal 7 site that, for now, must remain in 7. Its last version was 7.78. Each time I attempt to go to 7.98, or even a lower version such as 7.95/7.96, the theme breaks partially (not totally). It should be noted that the theme is a modified version of "MAYO" but that works as intended in 7.78. The entire Sites folder is being used in the update as-is, not changes. The following errors are displayed upon running Update, but do not also appear upon Update or Status Report in 7.78:

I am preparing to migrate a fairly complex site from Drupal 9.5.9 to D10.  I think I'm understanding most of what I need for this project, except for one thing, the documentation about migrating dependencies on core JQuery UI libraries.  I'm just not confident that my diagnostics are doing what they need to.

Specifically,  did (and found the following):

I am using upgrade_status to prepare my d9.5.9 site to upgrade to d10.  I'm getting close, but noticed a curious set of cases, and I hoped others might be able to clarify.

At the bottom of the upgrade_status report is the nice green listing of modules that are compatible with Drupal 10.  However, just for fun, I performed a scan on these modules and that reported that many of them still show "problems" according to upgrade_status.  

New here, but I've taken over site management for a D8.5.15 site running on PHP 5.5.38.

Is there a guide to upgrading, or do I need to be at least PHP 7+ to upgrade further?

Thanks, much.

After updated to the latest 7 update my update module feature is broken.

I cant update any module. When I attempt to update I get errors for example:

    Error installing / updating
    File Transfer failed, reason: Cannot copy /tmp/update-extraction-3516e48b/module_missing_message_fixer/README.txt to /home/user845/public_html/domainexample.com/sites/all/modules/module_missing_message_fixer/README.txt.

Any ideas ?

I've been working on migrating our Drupal 7 site to a Drupal 10 site the last 6 months. I did an initial complete migration, using the migration modules and processes provided and documented by Drupal, back in February 2023.