I have different thoughts on this.
Both Omnivision camera chips (1st OV8865 vs. 2nd OV12870) can provide videos in full HD although the new one could catch uhd videos (relying on their original datasheet from the manufacturer). I wasn´t able to find the corresponding datasheet for the xx70, only for xx90, but 12MP looks like uhd and is usually only a matter of a different frame rate. So xx90 could go up to amazing 270fps for HD and 60fps for UHD.
I assume your xx70 type would just go a bit slower nevertheless still faster than the first OV8865 chip though.
This then could result in a higher data rate being processed which again would cause a higher heat development.
I don´t know for sure were the limitation is made as to my knowledge FP2 is only providing full HD video streams, not higher.
So if the new image chip itself is limited by its internal software settings (there are 64 bytes customer memory reserved) it wouldn´t deliver higher video shots than full HD and not more data to the SoC. Then there should not be a noticeable difference to what we had with the first module.
If it´s however going full throttle pushing all the data to the SoC which therefore limits it to full HD there is a chance for a temperature rise as all the data is being processed but saved only in a full HD sized video file. These are internals only FP can know.
For still images there is no problem as they are taken snap-by-snap and afaik there is still no series function available.
That´s a good idea, probably needs root access though as I assume.
Yes, that´s what can be expected. There is a difference though between playing back videos and recording videos. For playback only the SoC has work to do. While recording both, the camera chip and SoC are involved, the latter has even more to do as video data processing (compression) is running as well.
Someone could put it that way.
But remember what @Dr_Cool has written. Heat development is normal and cannot fully be avoided and mobiles usually do not have any fans installed for active cooling such as conventional computers mostly have.
There´s only passive cooling done which is provided by the “frame” functioning as a heat sink.
I see different options to change this without any customer having to take approaches (or a risk of burned fingertips 
) as you have done like limiting the clock speed or any other action.
Since the display only goes to full HD and the software also limits videos to full HD what for take video streams with much higher resolution. This could be limited accordingly directly in the camera chip (remember 64 bytes for the customer). Then the SoC would also be relieved a bit. (only picky users would now rise their fingers as there generally is a difference).
A frame rate limit would help too as it is no action cam. Anything up to 60fps should be acceptable for all purpose (always considering the data stream between camera chip and SoC)
I know my phone only partially from the inside. Maybe there is more room for advancing the heat transport. I am not sure if the SoC case in only touching the frame or if there is a thermal heat pad clued between them. If not, this could help. (almost anyone could apply such by himself btw.)
At last that´s the design of FP and I think their engineers considered such things already. But we users are more in quantity and are making more experiences with the phone in every day life. So there is space for technical advancements.
the cpu