Time stops nearly everywhere, time only continues in a area around creatures of a certain intelligence. The worlds population is suddenly thrown into darkness as time ceases throughout the universe except near them, since light is no longer reaching them from outside their time bubble. They also find all electricity cuts out if the power source is outside their time field. Some people suffocate because they stay in one place to long, exhausting the air in their bubble. People can no longer rely on the sun for agriculture. Starvation is a serious threat, but so is asphyxiation.
It is probably a given that possibly the majority of the worlds population will die. I am wondering whether it is possible for humanity to survive in the long term, and what civilization the remnants of humanity could create. I'm not quite sure what is going to happen with all those animals that are smart enough to create time bubbles in the wild and I don't know how much that affects things. It does occur to me that the intelligent animals might not die off for quite a while as they would migrate to find new food. Although one thing to be considered is that I don't think there are any animals that are well adapted for total darkness that also pass the mirror test, so actually the animals that would generate time bubbles might die off quite quickly.
As for what methods humanity would employ to survive, and I suppose any permanent settlements would probably need CO2 scrubbers. People could move around nomadically since time doesn't pass outside time bubbles so if no-one has been somewhere since time stopped it will be exactly as it was at that second.
The only places time continues are around creatures that have incredibly high intelligence including great apes plus some other primates, cetaceans, many corvidae, elephants, parrots, pigs and a few other mammals and birds. A creature can only make time pass near itself if it individually meets the required intelligence bar, as thus human infants wont make time pass in their presence until they could pass the mirror test, this usually occurs at around 18 months.
The time field around a creature, in which time passes extends from every part of their body a distance equal to twice the distance between the furthest 2 points of that organisms body, this only counts living tissue (so not hair) that is a part of its body. This distance obviously changes: for instance if a person simply held up their arm then the distance between the furthest 2 points of their body increased: from being from their foot to their scalp, to being from their foot to their hand.
As for what happens to object partly on one side of the literal temporal barrier and partly on the other side (assuming the barrier either has no width or is only a planck length thick), I'm assuming through hand-waving that things straddling the temporal line aren't just cut in half and are still sort of attached to the portion on the other side should you pull on it.
EDIT A object can't be passed through a stationary time barrier. Once a object reached the edge and a tiny part of it has crossed the barrier it would be unable to move and would block the rest of the object from getting through. A object part outside the barrier and part inside could not be pulled inside either because the part of the object outside the barrier can't move but is still attached to the part inside the barrier. A object normally wouldn't be able to touch the edge of a stationary time barrier anyway because there will be a layer of air molecules trapped partly on both sides of the barrier. Pulling a object that is partly outside the barrier could be done but you would be breaking the object by pulling it hard enough to rip it apart.
In this scenario everything still has a gravitational pull on everything else but objects outside time bubbles just don't react since that would require time. In addition I'm assuming that since from the perspective of people in time bubbles the earth is no longer moving that people don't just get flung at deadly speed as this would kill everyone in fractions of a second. In this scenario I'm assuming that anything that would kill everyone within a incredibly short time is somehow resolved, so this situation isn't enormously dull.
In terms of intelligence of a certain level being needed to make a time bubble, I am assuming that brain damage or certain mental disorders emerging may cause a creature to stop creating a time bubble but that these need to be reflected by neurological changes to have a effect, thus a creature that is unconscious or drugged will still generate a bubble.
Ok regarding heat buildup. I do definitely think now that heat buildup is one of the biggest problems in this scenario. I think the way the bubbles edge would handle heat is like this: there is a nearly indestructible layer of air straddling the plank-length edge of the bubble however even with a creature holding perfectly still the bubbles edge will move by molecules at a time because no creature could truly hold totally still. Most heat would indeed build up because it had no-where to go, there would only be one way that heat escapes and that would be through light mostly infrared. Light emitted within the bubble will hit the edge unlike solid object that would be held back by the layer of trapped air molecules, even if the edge of the bubble moved the light would just resume its movement away from the bubble hitting the edge again. Thus light would be basically the only way heat would escape the mostly closed system of the bubble.