My understanding is that the client's browser moderates all cookie interactions. So the D2L platform doesn't inspect or manage any cookies set by a 3rd party tool.
It sounds like you're interested in how authentication works between a 3rd party App and D2L. Authentication thru Valence uses tokens in the query string, rather than cookies. So your app needs to send the appropriate tokens, and then D2L will send back appropriate tokens. For example, your app sends an x_a and x_c value to identify itself as an App, and an x_target value when it contacts the LMS to request user tokens. Once the user logs in, D2L needs to send back all of those original values (using the x_target to identify where to send the response) and also sends back the user tokens as x_b and x_d values. I'd encourage you to check out the Authentication topic to see this process mapped out in detail.
Does this help answer your questions?
~S-BB