As a practioner-student of Advaita Vedanta I'm answering your question on basis of my sampradaya's (sect's) interpretation of the topic.
Acharya Sankar Bhagwatpada has, in accordance with other Advaita Vedantin texts, always mentioned only the three states of mind (अवस्थात्रय/Avasthatraya) id est the Jagrata, Swapna and Susupti. Turiyavastha is recognized as the fourth state thereof but it is not counted as a true state of mind because it is by its innate nature beyond the domain of mind. Turiyavastha is therefore the state of detachment from the physically and mentally perceived world when the practitioner is completely detached from the mind (as well as from the body which is merely a manifest/function of mind). Turiyateetavastha is also the same as Turiyavastha except that it is final and irreversible inasmuch that the person having attained this state never comes back to his or her mortal person. Sriyogvasisthmaharamayan describes two classes of Aatmgyanin /self-realized persons : Jeevanmukta and Videhmukta. The former has attained Nirvana whilst living in his mortal form of flesh and blood and the latter having also given up his or her mortal form never to be reincarnated again. Therefore, tureeyavastha is the attribute of a Jeevanmukta and Tureeyateetavastha is that of the Videhmukta.
[It is relevant to mention here that the concept of mind is not the same in Vedic philosophies as understood through its English interpretation. For a better understanding thereof cf. अंतःकरण चातुष्टय / antah karan chatustya / the four inner facilities per Vedanta. It is also noteworthy that other Vedic philosophies like Samkhya and Yog recognize only three inner inner faculties]
Also, the fifth state as mentioned by you rarely, if ever, occurs in scriptures being described as an avastha / state and as such the term Tureeyateeta is often used merely to denote the formless Brahman, without the term 'avastha' adjuct thereto.
Now, let us deal with the core part of your question. The perception of time a subset and domain of the 4 mental faculties. It may or may not occur in either of these 5 states because the 4 inner faculties shall always remain the subset of True Nature of Being id est Brahman.
In susupti avastha the perception of time occurs in case of plants (who are said to be in susupti) but not in animals who are in comatose state due to medical condition (also said to be in sushupti). It also does not occurs in case of inanimate objects such as rocks et cetera (recognized as subset of same Consciousness as us albeit in a state of ghan-Sushupti)
Perception of time occur in tureeyavastha when the Nirvana-attained individual is engaged in his warranted conduct but may or may not occur in his state of samadhi.
In tureeyavastha the practioner is dissolved in the all-pervading True Self and witnesses all the three times (past, present & future) of all the infinite universes togather at once. In such state He becomes Time itself.
In order to be able to completely appreciate this theory merely academic efforts shall not suffice. It is paradoxical and described as beyond the domain of speech and mind, therefore it may only be understood through tradionally prescribed practice under a bona-fide and unbroken lineage of preceptorship.
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