What is the best time to schedule regular updates on inhouse production server?

Why not look at the concurrent usage of your system historically & determine what times of the day usage is at its lowest? Then stick your change right in the middle of that low usage period.

When working out how long the change will take include pre/post implementation testing and production verification testing. In addition work out how long the change will take to roll back if any testing fails.

IMHO your 'first users' shouldn't be guinea pigs. Having live users basically production verification test your changes is not a good thing. It destroys the end users confidence & the unexpected outcomes can mess up production which means not only do you have to roll back the change, but also roll back any 'damage' the change may have caused.

I don't know of any research papers, but take a look at any IT Service Management framework (ITSM) such as ITIL, you will find lots of standards & best practice on software release management. All systems are different so the extent of how many of the practices you adopt, and the formality, depends. ITSM standards have big systems in mind.

This is entirely dependent on the nature of the business. Some offices are 9-5 five days a week. Other businesses are 24 hours a day, 365 days a year. Other factors such as staff and resource availability play a significant role. No research paper could comprehensively cover every possible schedule or eventuality.

Ultimately, the management of the company or department in concert with IT management have to determine what is best.

The key to success is communicating with users when the down time is scheduled to start, how long it's expected to last, any preparation required of the users and what they can expect as a result of success or failure. A big part of that is meeting the expectations you set.

In the end, nothing is etched in stone. If the process doesn't work then make adjustments. Your flexibility and adaptability will be appreciated.

By performing maintenance and update procedures on test equipment beforehand when possible, you will be better prepared when it comes time to implement them on production systems.

I work at an ISP and in my experience, most of the people I would consider heavy hitter system administrators choose Friday evenings on holiday weekends to do their major network overhauls. That gives them an extra 24 hours to test and if necessary roll back their changes. However, to a large degree this is entirely dependent on the nature and habits of your users.