There is a couple schools of thought when it comes to leaving the eggs... some feel that if you leave the eggs it will encourage nesting, others feel that if you remove the eggs, the bird will keep laying until she get to her full clutch... I think most people choose to leave them though (unless of course the egg is broken then you remove it asap due to bacteria)

If they are infertile I think you can leave them until she looses interest in them (I *think* that's ~3-5 weeks), or if they get damaged.
Do you know if it's a male/female pair for sure? You can candle eggs (someone else can explain how to do that) or just wait for them to hatch...
There shouldn't be a nestbox if you are not prepared to raise chicks.

The nestbox encourages egg laying so by providing the proper environment will cause the female to lay eggs. If you don't want chicks, then don't provide the box... or any nesting materials.
Since the female is laying you'll want to provide extra calcium in her diet (cuttlebone, broccoli, kale etc.) and decrease her amount of daylight.
Hope that helps, I'm sure someone else will be around shortly to clarify some things since I'm not a professional!

Congrats on the new Lovies, looking forward to some pictures!