Noise at start and splashing(?) sound

Hello! New in here and also bought my first used Prius -08 a week ago. Too bad it had been unused so long that 12v battery has to be replaced, it was next to dead. But cause I bought car from dealer, they promised to give me a new one without cost.

And yes..Prius makes all kinds of weird and almost funny noises; brake pump, electric motor hight pitch on braking and on ev mode.

I have 4 bare with me-questions:
When I start the car, before the ICE goes on, I hear a noise way back from drivers seat. Sounds like from back seat/trunk area. Sound is kind of pumping, low noise and goes on for maybe 3-5 seconds. Is it the gas pump?
I have heard gas pumps before in cars on start and these is quite similar but louder

2: when I start engine cold(its around 5-10celsius now here on Finland, about 41-50 F), the ICE idles little uneven. RPM seems to be little waving, not much but audible up and down. I aint no expert but the rpm change is not big, maybe changing max 100-200 rounds up and down. Is this normal? My earlier car started cold in higher rpm but did not wave up and down.

3: also to do with earlier question; when ICE is still cold( not freezing anyway) the first couple of stops at lights I very clearly feel a shudder. Keeping the brake pedal down and waiting on red light, the car shakes back and forth so that I feel it and even think someone standing outside car might even notice car trembling. Also ICE sounds abnormal at the same time; its not smooth sounding, kind of like misfiring, uneven sounding. Feels like gasoline is just about to go empty or car is stalling. Is this normal in "cold" weather? Heck, its not even below zero C here and car sounds like that.. waiting for winter and it gets -25 celcius..

4: last and least; when stopping car from motion, I can hear splashing sound from back. My gastank is now full so that must be it? Sounds like typical full tank noise; car goes stop the splash comes and gentles in few seconds.

 

Also a little note for question 3. The shudder back and forth is more feelable(?) when the car is facing a hill. So braking and waiting in upward arching road, the trembling is more feelable than on straight ground.. Thanks.

 

Ok a little correction to the previous message. When star key is placed in and the moment i press brake pedal BEFORE pushing power-button happens two things: brake accumulator(?) makes a sound of fast clicking which I believe is normal charging sound and happens also during driving sometimes. But at the same time some pumping sound is heard from FRONT of car. So not from back as I said before.

The splashing sound is pretty sure from fuel tank in sudden stops as liquid moves around in tank.

Note that 2nd gen Prius cars sold in Europe I think do not have the big coolant tank where warm coolant is pumped when you turn engine off and from where coolant is pumped back in start. So this sound is something else. Maybe I have to make a video of it..

 

I'll check that, but I think it's gas tank. I haven't yet seen any moister or water in back. But I'll check better again.
Car is driven 77.000km = round 48.000 miles .

 

Wow thank for expert reply. Now the temperatures have risen a little, around 50-60 F days, night are still about 40 or less - so car seems to be little less rough but I still feel the vibrations while waiting in stops after the start. Driven maybe few minutes, the idle goes calm and no vibrations(well nothing out of normal sensation).

When I start the car in mornings, pushing the brake pedal(even before key in place), I hear brake accumulator electric sound, a buzz - also at the same time some clunks from front for 2-5 seconds. It's not bad noise, just weird.

Also when I turn A/C on or off, these is a moan from just behind dashboard. I think some a/c vent keeps that noise. Any idea how to silence it, maybe greasing hinges or somethng?

 

See my video taken this morning. Car has not been driven around 14 hours.