Will creating the DC (and by extension, domain Y) on domain X cause any issues?
You cannot have a domain controller for one domain as a member of a different domain. Domain controllers, in effect, are the domain for which they're on.
Will these two domains conflict with each other if I don't initiate a trust relationship between the two?
The domains will effectively not be able to talk to each other, unless and until you establish a trust relationship.
Frankly, these are pretty basic questions that you have, and if I were you, I'd lean on consultant, or a more senior colleague with relevant experience. With your level of knowledge, you're begging for a disaster trying to set this up (and do a migration) yourself.
At a very minimum, you need to educate yourself on Active Directory. You could start with the excellent thread by @MDMarra that we have here.