For your first issue… have the host disable the old shared hosting version. Basically the only place your data should be “active” is the new server. Next… make sure your conf_global.php reflects the correct URL that you want (including http vs https prefix). After that is correct, make sure you clear your system cache in ACP support section.
For your second issue, that path mentioned does not look like a default IPB setup. Are you using a 3rd party mail service like Sendgrid? If so, you might need to fix your setup there. If you’re using the default phpmail, you might have some sort of third party resource that is acting up.