Is it permissible to supplicate for an expedited answer to a request in this worldly life, or is that considered overstepping and hastening in supplication?

There is no harm in asking Allah Almighty to hasten the answer in this world, as long as the matter is something permissible to supplicate for according to Sharia. It is better to entrust the matter to Allah, as delaying the answer until the Hereafter might be better.