Is there a way that I can install (hosted) Lync on our terminal server and when a user logs on, it automatically inserts their email address and the correct manual SIP settings so I don't have to log on to each profile manually?
If you're speaking about Lync client the answer is yes, you can: you users can configure Lync client to remember credentials and to startup at login. Here a link that provide you more information about supported features on Remote Desktop.
By "hosted", I'm assuming you mean Lync is not actually deployed in your network, but instead you are talking about Lync Online; also, your is not even a hybrid deployment, otherwise the sign-in address would be automatically populated for all Lync-enabled users (if this is not your case and you really have a standard on-premises Lync deployment, then something must be very wrong for this to not automatically work).
In this scenario, the Lync (or Skype for Business) client tries to get the user's SIP address from Active Directory, but it can't find it, so it just asks the user; this behavior is the same regardless of which computer you're using (i.e. the fact that you are using a terminal server isn't relevant at all).
The solution is to populate the correct Active Directory attribute for Lync-enabled users, so that the Lync client can automatically know which SIP address it should use; said attribute is named
Also, about SIP settings: if your DNS is configured correctly (as per Microsoft instructions), they will automatically be detected by the Lync client when signing in; no manual configuration is required. If you put in a correct sign-in address and password and the client still asks for the SIP settings, then either the client is configured for manual settings (but it shouldn't, automatic configuration is the default) or something is wrong in your DNS configuration.