Question about Wood's MDK21

No. Here from the manual:
Memory function: The dehumidifier remembers the set values as long as the mains plug is plugged in. If the dehumidifier is switched off and the mains plug is removed from the socket - the dehumidifier starts in standard mode when it is switched on again.
After every one hour, the unit starts up and measures the humidity. If the value is above the programmed value, dehumidification starts until the value falls below the threshold value. If or as soon as the humidity is below the threshold, the unit goes into standby again for one hour.

No, after any restart it will start with factory setting I do not know why you want to put the timer but the setting is that every one our works for 2 min and measure the humidity if the number is less that your set number it will stop till next hour.

Is the hose supplied or if not, what type of hose is required (inner diameter should be decisive - thank you very much)?

I can confirm that it only keeps the settings as long as it was switched on before disconnecting the power supply. I use a Hombli smart plug for the control, with which I can also switch it on and off remotely. After switching on, the fan runs for 5 minutes before the compressor starts as compressor protection. Hombli Smart Swiss Socket 2+1

I plugged it back in after a week, it started up again with the previous setting. I don't use the timer on the device. If you use an external timer it should be possible

Yes, it is ideal for an apartment/house. it simply continues to function the way it was plugged in.