It depends on the queries being run. The optimizer will take into account when an index is unique and it can improve performance, but it really does depend on the query. It might not affect things at all.
It will slow down inserts slightly, but probably not enough to notice. It doesn’t have to check everything single record against every other record. It’s going to use the index itself for the check which will be quite fast.
In general, if you’ve determined that an index will help and the data being indexed is going to be unique, I’d say go ahead and make it a unique index. It won’t cost much and you will receive benefits.