As asked in the question, what are the characteristics by which a thing is classified as 'living being' or 'non-living' ? How is a lifeform defined in Buddhism?
A living being 'breathes'. The Pali word is 'pāṇa'. The Pali states:
Idha, cunda, ekacco pāṇātipātī hoti luddo lohitapāṇi hatapahate niviṭṭho adayāpanno sabba pāṇabhūtesu.
There is the case where a certain person takes life, is a hunter, bloody-handed, devoted to killing & slaying, showing no mercy to
Cunda Kammaraputta Sutta
This is how a 'living being' is defined.
I mean, like how humans consider themselves as living beings due to
various facts such as: their experience in this world due to
consciousness, their actions (kamma), etc.
A "living being" (pāṇabhūtesu) is not the same as "a being" ("satta"). A "satta" is merely an idea or view. Please refer to SN 23.2 and SN 5.10 for an explanation.
if a living being (pāṇabhūtesu) perform kamma (actions), they develop ideas or view of 'self', which is the meaning of 'a being' ('satta').
Note: Not all living beings perform kamma. Only homo sapiens & possibly higher life forms perform kamma.
If lifeform is defined as something that has consciousness and capable of producing kamma, what things are conscious?
Most living beings have consciousness, even if it is very primitive. For example, a worm will wriggle & try to escape if you touch it. That a worm senses sense stimuli would classify it as a conscious living being.
Where as a tree may breathe (carbon dioxide) however a tree is probably not conscious because it cannot sense & react to sense stimuli.
Is a plant a living being/conscious?
I answered this question before I read it. A plant is probably not a living being in the sense of the word "pāṇabhūtesu" because killing plants does not fall under the precept about killing.
However, plants are obviously considered to be life forms.
Is a stone living being/conscious?
Of course not because as stone does not breath or have consciousness.
Is the universe conscious? Is everything a lifeform in some way?
That question is too deep & outside the scope of Buddhism. There are forces in the universe such as homeostasis & gravity; which infer some degree of communication but this is outside of the scope of Buddhism because Buddhism is only about ending suffering.
Buddhism states life (vitality) is dependent on heat & heat is dependent on vitality (MN 43). So what exactly is "life"? How long or short is a piece of string?
Is rebirth possible only as a lifeform? (I mean, by rebirth, just a transfer of consciousness/experience subject to different conditions.
Please correct me if I'm wrong)
In Buddhism, the word 'birth' ('jati') refers to the arising of 'self-view' or egoism. Please refer SN 22.81, for example, or SN 12.2, which defines 'birth' as the production of the view of 'a being' ('satta'). 'Re-birth' is psychological rather than physical.
There is the case where an uninstructed...person — assumes form to be a self. That assumption is a fabrication. Now what is the cause, what is the origination, what is the birth, what is the coming-into-existence of that fabrication? To an uninstructed, run-of-the-mill person, touched by that which is felt born of contact with ignorance, craving arises. That fabrication is born of that. SN 22.81
Can our next rebirth be as a stone/whatever, other than the 5 predefined possibilities (human, god, ghost, animal, hell) in Buddhism?
If you are humane, moral & non-violent, this is a human rebirth. If you are extraordinarily loving, wealthy or powerful, this is a godly rebirth. If you are addicted to craving, this is a ghost rebirth. If you are immoral, violent & unthinking, this is an animal rebirth. If you are suffering or angry, this is a hell rebirth. In one single hour, you can reborn many times, whenever the mind grasps or mistakes different emotions as 'self'.
The next time the mind thinks the thought "I", "me", "mine", 'self", "him", "her", "them","us", "your", "you" or "our", this is the next rebirth. The Buddha taught each new birth brings suffering.