Base path and user picture uploading
Firstly, it seems that the new server does not allow clean urls or it needs some configuration. Check the documentation on Drupal.org: Step 6: Configure clean URLs

Then, clean all caches and run cron. You will then see all image paths as /new/sites/sites/default/files/...

That's because the new site is set on folder ROOT/new or am I wrong? So there is no error, you have to set the domain to look at this subfolder.

If you don't have permissions to do this in your hosting or there are no options like Add-On domains you can follow the discussion at Drupal installed in subdirectory but made to appear in root according to your hosting enviiroment.