minimum lease period

If you can find a replacement tenant to assign or sublet to, then you can ask the landlord to allow you assign or sublet, If they refuse, you can end the tenancy. It is better to assign than sublet from your point of view.

This is incorrect, a Part 4 tenancy adds to your rights but doesn't overwrite the written fixed-term lease. Otherwise no-one would bother with a fixed-term lease at all!

You gain Part 4 tenancy rights, meaning you can stay in the property for a further 5 and a 1/2 years (subject to certain exceptions). Part 4 extends your rights, but it runs alongside a fixed-term lease, it doesn't overwrite one. You can use Part 4 to extend your stay but not to terminate before the fixed-term applies. At the end of a fixed-term lease your landlord cannot make you sign another one as you are already covered by the Part 4 tenancy.

A fixed term tenancy can only be ended if the landlord breaches their responsibilities, or if they refuse to allow you to sublet or assign the tenancy.

https://www.rtb.ie/ending-a-tenancy/ending-a-fixed-term-tenancy

What happens to specific provisions in the lease under those circumstances? E.g. if the lease states no pets, or no smoking indoors or similar and then you don’t sign a new lease when the fixed term is up? Do those conditions still apply?

It is provided in the lease that the same conditions continue with the lease turning into a month to month lease.