You can copy and paste an actual hyperlink into the Confirmation Notes that will be sent by email. The same details are also sent if a Reminder email is being sent. You are only able to use a hyperlink in the field, not an embedded link. An example of an embedded link is when you have text in the message like Click Here where the word Here contains the embedded hyperlink.
If it is helpful, there are online 'wiki' entries or other web resources available for how to shorten a URL/web address which might help to save some space in Confirmation Notes.
For system emails, if there are hyperlinks being displayed, the colors are based on the Color Scheme your team has set under the System Settings menu. The Primary color specifically dictates how those hyperlinks are displayed. Any staff with access to the System Settings menu can adjust the colors being used (System > System Settings).