Hiya.
as far as I am aware, ezy do not charge for desk check-in. I am not familiar with Luton Airport though, so hopefully someone else can help you there.
Info on hand and hold luggage restrictions can be found on easy website, or in your t&c's of carriage, but basically if you booked a hold bag for one person, that must be 20kg max, if you book another bag for the same person, the weight allowance does not increase, still 20kg.
I you, however, paid for 1 hold bag for two pax, the allowed weight is 32kg for that one bag. Your booking information page should show this.
ONE (strictly) item of hand luggage is allowed per person, size 20 by 40 by 50 cm, it must fit in the cage at check-in. There is no weight restriction on that, as long as you are able to lift it into the overhad locker without help.
Hope that helps, enjoy your hols!
B