1.
Use existing domain and only change the name-servers. (Or buy your domain separately, for example, from name**cheap if you don't have any existing non-important domain to move there). IMO, don't let them buy a new domain for you, or don't you do a Registrar Transfer so soon... just do a name-server change initially as mentioned above so that your domain re-transfer later on will not be unexpectedly more expensive or difficult in case you don't like the new hosting service...
You can always use the cheap hosting ($1 is cheap!) as your secondary hosting service to do some important testing, and if you don't like their overall hosting service then 1st remove your domain to your primary / another host by changing the name-servers back to your primary or a new host. No doubt you'll also take a full backup frequently and transfer the backups alongwith your domain name if need arises.
3.
Recommended: keep your existing premier / quality hosting service when buying the $1 hosting service. You will of course have some more time to pay the next subscription to your 1st host, so use that time to test this $1 host thoroughly.
IMO, there is also no harm in continuing with your primary hosting service for one more year, and if this new service is good then decide whether to move on to it entirely whenever you think its ok to do so. Remember, this $1 new hosting service subscription will turn to around $70+ from the next year, but you'll have sufficient time to test it, esp their uptime, security, cust service, etc. Of course, with this promotion, expect them to cram hundreds / thousands of sites more on each server, so the overall server (and service) performance wouldn't be as good later on.
My 2 cents...