Re 1
Check how much power you have on a given phase of power consumption (if there is no situation after turning on the receiver / computer or vacuum cleaner, the total instantaneous power / sum of power of all receivers that can normally operate with a given fuse / - the easiest way is to get to the fuse box and by turning it off or on, track what you have connected to a given power line).
Then calculate the power of everything that is connected there: voltage, result in Amperes, add +25% to this result (correction just in case, e.g. connectors or start of receivers) if the value is close to the fuse amperage (about 16 A / about 3kW in a thicker round ) then you have an overload of the power line when the device starts up (you can circumvent this with toys so that they turn on the receiver at the zero point, but it's too much to mess around with supplies in the network and you need a certificate to get it so that there are no problems in case of an accident, self-made so drops out).
As you have 16A fuses on a given phase and you pull > 2.5 kW, it is possible that the overload at the time of switching on the receiver (temporary increased power consumption for charging electrolytes, etc. start-up - it is normal that during start-up the rated power of the receiver is exceeded by some percentage). The most sensible thing would then be to switch something to another phase to reduce the power consumption from a given phase.
Since both the power supply and another receiver cause overloads, it is normal (exceeding the current and therefore blowing the overcurrent fuse) and correct because this is how the protection works so that, for example, there is no fire due to a short circuit in the electrical installation.
It is possible that something like this happens: the sine wave goes up and down, you turn it on and there is a temporary break above the level ('pin' in the power supply) as a result of connecting another receiver, which in turn triggers the fuse.
Re 2
Something like a soft start would be useful, but I doubt if computer power supplies etc. will like it and if it will not be damaged due to the soft start system (read: the power supply could not survive it).
Try to give a stronger strip and if it doesn't work, relieve the power line (phase).
Added after 2 [minutes]: km4 wrote: connect the power supply without the power strip and see if you blow the fuse. Is it a good idea?
I have the same vacuum cleaner in my room. In my opinion, the fuse is too sensitive, replacement should help, although let others speak.
Reaction time, but I doubt you would get something legal and certified, it's best to talk to an electrician, probably SEP'owszczyzna - that is, the response time of the fuse, and the pulse width of the "pin" and safety in the event that this 'pin' was the result of an electric shock of the human factor (i.e. slower a fuse can kill [allow an electric shock for too long] and a too fast one will blow every single pin).