I have previously worked in hotel management, and these days travel a lot. For the past 7 years I have spent 10-15 nights in hotels per month. These hotels have been both domestic and international, corporate and boutique, for both business and pleasure.
Room deposits are always run at the beginning of the stay (no point in taking a deposit otherwise!). If I haven't pre-paid, actual room charge and incidentals are run together, upon check out. Deposit is then returned within 5 working days. This seems to be the case for 99% of hotels.
I've never had or heard of guest bills being put under the door? This seems very old fashioned. Closest I've come would be receiving a copy via email, and these statements are generally sent after check out (perhaps twice in this entire time have I received the statement before). I also always check out at the front desk so I can obtain a physical copy of the receipt. I can't say I know anyone who doesn't present at front desk upon check out regardless of whether they want the receipt or not.
May I ask what kind of hotel you work at and in which country? I'm very interested as I find your comments to be the complete opposite to my experiences.