The rules do not directly address this. However, we can extrapolate what would happen based on existing rules (referring here to Savage Worlds Deluxe).
Would the Gizmo remain working, as if the weird scientist had given them to the players (if they loot it)?
I see no reason why not. The rules already provide for a way to let other players use them
Sharing: An inventor can hand his device over to a companion to use. The device works the same for the companion as it does for the owner.
There is no reference to the inventor needing to be present for it to continue to work (which I suppose means they could be alive, but in a whole other plane of existence), thus I believe that they still function.
Would it still regenerate Power Points?
Depends on the trappings (though I would say "no" for game balance). Here is the description of recharging power points.
Recharging: Devices regain Power Points just like a character, at the rate of one point per hour. Recharging may represent the device being plugged into an outlet, gathering solar rays, or even fritzing out for a while until it miraculously just starts working again.
All of these trappings imply that they recharge without the aid of the inventor. In contrast, settings like Deadlands describe some alternative source of charging that could not happen without the mad scientist:
(Minor spoilers from the Deadlands Reloaded Marshal's Guide below)
As you may have sussed out from the rumor and innuendo in the Deadlands Player's Guide, Marshal, most inventions of the new science are the result of manitous "inspiring" mad scientists. Though most scientists don't know it or won't believe it, these evil spirits whisper in their ears while they sleep, in many cases suggesting designs that would not work at all were it not for the supernatural properties of ghost rock.
Since the source of the weird scientist's devices are supernatural beings and supernatural ghost rock, it would presumably cease to work if the conduit they are using no longer exists.
That said, I think that there is a balance issue if you allow powers to recharge on their own in the absence of an inventor (this would be a problem for sharing too). If a Legendary rank Weird Scientist shows up with 9 powers each with 25 Power Points and dies in the first session, it might be a bit unbalanced to have remaining players have them accessible and recharging for players that don't actually contain the Edges that created them.
For balance, my ruling as a GM would be that devices don't recharge if there is nobody in the party that has an Edge that created it.
Would it simply stop working, in a way similar to a Gizmo that was replaced by the Weird Scientist after they lost ist?
I assume that you are referring to these rules about replacement devices:
Losing Devices: If an inventor’s gizmo is taken away, lost, or destroyed, he can make another in 2d6 hours as long as he has access to a workshop and some basic components. This requires a Repair roll. A raise halves the time needed to create the device.
If the original should somehow be returned, it no longer functions (the character doesn’t get a “free” duplicate in this way).
It's clear that the device no longer works after the replacement is returned. Nothing is said about before then, so I would think that the Sharing rules apply.
Given this, if the character dies, then gets resurrected and builds another one, his original won't work any more if he gets it back.
Devices stop working on a roll of a 1.
Malfunction: Weird science devices are never perfect technology. They often suffer from spectacular and deadly malfunctions. If a gadgeteer uses a device and rolls a 1 on the skill die used to activate the gizmo, it has malfunctioned in some way and does not work. Draw a card and consult the Malfunction Table below:
The results on the table involve it either exploding or being damaged. While the text is a bit unclear of if the gadgeteer must be the one to repair it in the latter case, I think the intention is definitely that it's them (ultimately it's the GM's call though).