I mean, can I just relax, and slowly play as fast as I want, or there is some kind of time-quests, which will abandon after some ingame time, so I need to rush through a game using saves and time-management?

I'm on day 4 and so far there were no time limits... In fact, you can't end day 1 without doing a certain quest... I haven't finished the game yet though, so I can't say for sure.

Remember that time only passes when you talk to people/interact with objects, and every night the clock stops at 2.00(you have to go to sleep to start the next day).

The developer posted this on Steam:

"There are no timed quests in Disco Elysium. The clock symbol next to a task in your journal does not mean it’s “timed”. That is: do it by a certain time, or you will never get to do it. Rather, it means this task can only be done at after a certain time of day: after 21.00, after 23.00 etc. The time is always specified in the task’s description.

For example: the “end of 1st day Kim debrief” task can be completed by clicking on your room door in the Whirling In Rags at the end of the 1st day, after 21.00. Kim will say "hey, let’s talk about the day before". (NOTE: you need to first have paid Garte for the damages for this.)

There are other clock-symbol tasks too. To track down the Smoker on the Balcony you need to go to his apartment after 22.00, as the task description specifies. But remember: if you don’t make it to the meeting after 22.00 – if you forgot or had something else to do -- you can always go again tomorrow after 22.00.

The tasks will not be gone tomorrow. They are there, in your Journal when you wake up next morning. And morning after that and so on.

Only doing the debrief with Kim is special in that way. It’s not available tomorrow, because it doesn’t need to be: it’s unavoidable at the end of day 1. Kim is very insistent :)

Hope this helps!"