AC Short-circuit Protection
My favourite relay for this type of application is the NAIS / Aromat JS1 family.  This is a form C relay with the NO contacts good for 10A and the NC contacts good for 8A.  We have used several hundred thousand of these relays over the past 20 years or so driving many different loads, including water circulating pumps for hot water heating systems.  No relay failures that weren't caused by installer wiring errors.

You don't need a fuse if everything connected to the AC Mains supply in your circuit is designed to handle the full current rating of the branch circuit that the unit is tied to.  That means wide traces on the PCB.

Most of our boards don't have fuses or any other form of over-current protection on the AC Mains side of things.  The low-voltage stuff in our HVAC boards uses a polyswitch self-resetting device to protect the power transformer should someone make a wiring error that would cause the power transformer to self-destruct.  We use certified Class 2 transformers that have permanent (one-time) over-current and over-temperature cutouts.  The polyswitch saves the transformer from dying because of a silly wiring mistake.

We use a larger relay when needed: American Zettler AZ2150 (form C) or AZ2150A (form A).  We use those for controlling, among other things, 3/4 HP induction motors used for fans and blowers.