Couple of ways, either use light weight blocks or stud board and skim.
Whichever way make sure you install a vent, and fit a rain guard on the chimney pot.
This will stop water getting in and keep stack aired.

Yes you can do this but its could practise to leave a ventilation grill on the front if the chimney is not capped, to prevent damp build up.

You can fill in the fireplace with thermalite blocks and bricks and then plaster over. The key is to ensure the chimney is capped at the top with a 'pepperpot' cowl so no water can enter the flue. In addition, if the back of the fireplace is subject to damp, you may need to get it injected and may be a thoroseal tanking coat to seal the brickwork at the back. Also if damp is present, use a rennovating plaster/skim coat such as Limelight. Apart from these there are no real issues.

your chepest option and the quickest is to build a stud frame in the opening set it back the thickness of the plaster board and then place the board onto the frame and plaster over it, altrnatively you could brick it up

There's no problem blocking-up / plastering a redundant fire place, but you should ensure the chimney top is closed-off to water ingress plus it is best practice to vent the flue space so as to avoid condensation forming.