You will need an overnight bag just in case, yes. Take comfy trousers with loose fitting waists, comfy tops, pyjamas or a nightie, extra underwear (again, loose fitting - if you don't have, sainsburys or tescos do some), slippers, dressing gown, wash stuff and something to do while waiting for things (books, etc). Would also reccomend - baby wipes, moisturiser, lip balm, sports top bottle for drinking, those juice drop things that you can add to water, light snacks for afterwards, hand cream. Don't forget to take small change, mobile phone, charger and any documents relating to surgery - admission letter, etc - and your usual medication (including all herbal and over the counter stuff that you use regularly).
With regards to underwear during surgery, it depends. I don't think you could wear knickers. I would think they would come too far up, and more to the point if you have knickers on when you wake up I don't think they'll help with pain levels. You'll have a flat pad underneath you for the duration of the surgery and afterwards too, so don't worry about periods or anything. I had my period during one of my gynae ops and it was dealt with sensitively - just a flat pad underneath me and then as soon as I was able a sanitary pad.
Piercing wise - in my hospital we ask for all jewellery, including all piercings, to be removed. I can almost certainly say that you will be asked to take your tongue, lip and smiley out. They'll need clear access to your mouth as you'll have tubes placed there during surgery to aid your breathing, and having piercings presents a major choking risk as well. Your nipple might be OK but I'd remove anyway - you'll have wires across your chest for monitoring during the op and I wouldn't fancy the risk of getting it tangled up and having another minor injury to add to the op! Ask the ward staff before you go - but I do think you'll be asked to remove everything that you can.
How long you will be asleep depends on what happens in theatre. You'll probably be out for at the very least, an hour. It depends if it's purely a diagnostic laparoscopy or if they are going to treat anything that they find. You'll 'wake up' in recovery but in all likelihood you won't feel totally awake until you're back on the ward. Most people continue to feel a bit drowsy for a good while afterwards.
I don't know if you can request to stay overnight. In all honesty if you were well enough, if your pain was controlled, if there were no complications, you'd be better off at home with your own home comforts. They won't allow you to leave unless they are absolutely certain of things being ok. You'll be given painkillers to take home if you are discharged on the day.
Filming and bill wise, I can't answer that - you will need to ask the hospital themselves.
Try not to panic. I have spent lots of time in hospital as a patient, and now work in one as an auxiliary nurse. If it makes you feel any better, I look after patients having operations every day - in my case, operations on the spine or on the brain (neurology). In the vast majority of cases, everything goes absolutely fine. Things only tend to get complicated when it's a massively complicated op in an older patient with dozens of health issues and previous operations. In a younger patient, with less health issues, no previous surgery, etc, the chances of things going wrong are very, very slim :) you'll be just fine, in other words :)