The mechanics in Derby Manager are working in favor of gameplay, often at the expense of realism. Here is a set of questions and answers to help you understand why certain things work the way they do.
I don't have/can't afford a license to show real club names. Also, seasons are going rather fast and any real players that would have been generated when the game started would now be long retired.
Manager names are generated randomly to avoid players naming their managers by popular real-world people or giving them funny/stupid names that would kill the verisimilitude of the game. It would look bad if we had 7 Pep Guardiolas, all of which are struggling in D divisions across the world.
As for the club names, those are given by biggest cities/regions/neighborhoods from each country. Colors are picked randomly to avoid insinuating that the club is stands for a real-world club named after that city. Manchester is simply Manchester, it's not United or City.
The matches start playing at the time indicated in the game, but it takes a while (up to 15 minutes) to run them all. It can also happen that there is a bug and all matches are late for a couple of hours.
This feature is not supported and never will be. There is only a handful of national-team games per season which would make it a boring endeavor. In addition, creating a whole new system for, at most, 32 people to play as national team coaches seems like a waste of time. Also, it would be very prone to cheating and manipulating the game...
It would be unfair. Top teams have more games per day which would mean that their injuries last longer. Having them counted in the number of matches a player will miss is the fairest option.
Derby Manager has action economy in place and loans would interfere with that. As a manager you get only a certain number of matches per season and you have to balance your team between training youngsters and winning games. Adding loans to that would allow you to offload training to someone else.
Considering playing matches often means an increase in player's value - the loanee would be at a disadvantage. He would increase the value of someone else's player. This gives him no incentive to actually loan a player. Even a potential short term benefit of winning more games is nullified when that player returns to his club and the loanee is left 1 key player short and no money to replace him.
Seasons are generated on 1st in the month. The first games are usually either 1st in the evening (European time) or fall on the 2nd. There are friendly matches on 29th, 30th and 31st while you wait for the new season to start.
On your finances screen there is an overview that doesn't include money from matches since that part is variable. Majority of clubs will lose money if you don't account for the money from matches (and some might even with it).
Another thing to take into account is that your total team value will also (hopefully) grow over time. So even if your finances stagnate, your team is probably moving forward and getting better by the day.