there are a couple of points in your post. The first is, it’s hard to say when a baptism is or isn’t conditional - if a person thinks they may have been baptised possibly, a some point but has no idea when or (more importantly) by whom it’s really just easier to start over - after all, Canon Law can’t envisage every possible situation and so some “Captain’s calls” are required.
Most people tend to want to be baptised in their home parish rather than at the cathedral (and Cathedral administrators are no doubt thankful for the reduction in the numbers they have to try and accommodate). At the end of they day, given that you’re supposed to be being baptised into the faith of the (parish) community this shouldn’t be a big deal (although obviously I can’t speak for your local bishop or parish priest).
As far as confirmation goes, it completes baptism (the two were originally intended to go together but got separated over time - at least in the Latin Church) which is why adults who are baptised are supposed to be confirmed immediately afterwards. Granted there are exceptions to this rule, as there are with most rules, although they normally only come up in the case of children. the best thing for you to do is discuss this with your parish priest since, ultimately, it’s his call.