Here is an incomplete answer just to get the ball rolling.
I haven't had the time to find "good" sources about the sound changes so this is just based on Wikipedia's "Phonological history of French" and my memory of things I've read. Take it with a grain of salt!
The change of Latin -ātum and ātem to French -é is a regular development. The vowel "a" in stressed open syllables was regularly raised in most circumstances in the development of French from Latin. You can see numerous other examples in "On a Morphologically Governed Vowel Alternation in French", by Franqois C. Dell and Elisabeth O. Selkir, although this article takes a synchronic rather than a diachronic view. I believe the form of the word "état" is due to influence from learned Latin or from other Romance languages; it should not be considered a model for French sound change laws.
An overview of relevant sound changes from Latin to French.
Latin vowel length lost (with quality changes), word-final "m" of accusatives is dropped (at least in polysyllabic words).
-tās, -tātem, -tātēs > *-tas, *-tatɛ, *-tates
-ātus, -ātum, -ātī, -ātōs > *-atos, *-ato, *-ati, *-atos
-āta, -ātam, -ātās > *-ata, *-ata, *-atas
(I'm leaving out -ātae because it's not clear that this form contributed to later French forms)
Latin intervocalic stops become lenited; according to Wiktionary "voiced stops and unvoiced fricatives become voiced fricatives (/ð/, /v/, /j/); unvoiced stops become voiced stops"
-tās, -tātem, -tātēs = *-tas, *-tadɛ, *-tades (assume there is a consonant before the first "t")
-ātus, -ātum, -ātī, -ātōs = *-ados, *-ado, *-adi, *-ados
-āta, -ātam, -ātās = *-ada, *-ada, *-adas
Change of "a" in stressed syllables: according to Wikipedia, "diphthongization of /e/, /o/, /a/ to /ei/, /ou/, /ae/ in stressed, open syllables, not followed by a palatal sound [...] (Later on, /ei/ > /oi/, /ou/ > /eu/, /ae/ > /e/; see below.)". I know there are attested spellings Old French spellings with "ae"; I don't know if this is the basis of Wikipedia's (apparently uncited) statement that stressed "a" diphthongized to /ae/, but in any case it's clear that it ended up being raised and fronted to some kind of e-like vowel.
Also "Loss of all vowels in unstressed, final syllables, except /a/" (Wikipedia).
Latin singleton intervocalic [t], which had earlier been lenited to [d], was by this point further lenited to a voiced fricative or approximant [ð], but when the vowel loss described in the previous sentence put it in word-final position, the reflex was a voiceless fricative [θ] (according to the Wikipedia talk page), and when vowel loss put it immediately before [s] it seems they coalesced to form the affricate [ts]. (I don't see this part explicitly described on the main Wikipedia page.) The presence of dental fricatives in 12th-century Old French, and their spelling as "d" and (word-finally) "t", is mentioned incidentally in "The evolution of English dental fricatives: variation and change", by Matiusz Jekiel (p. 51); this source mentions that digraphs "dh" and "th" were also used to represent these sounds.
-tās, -tātem, -tātēs = -tes, -teθ, -tets
-ātus, -ātum, -ātī, -ātōs = -ets, -eθ, -eθ, -ets
-āta, -ātam, -ātās = -eðə, -eðə, -eðəs
(Note: the vowel here might have been [ɛ] rather than [e]; I don't think they contrast)
In fact, the forms I give here for past participles correspond to the attested declension given for the past participle of 1st-declension verbs in An Introduction to Old French, by François Frédéric Roget (assuming [θ] is spelled with (word-final) "t", [ð] is spelled with (intervocalic) "d", and [ts] is spelled with (word-final) "z"):
Old French.
Singular.
Masc. Fem.
Subject.: portets (portez). portede.
Object.: portet. portede.
Plural.
Subject.: portet. portedes.
Object.: portez. portedes.
(p. 85)
After this, intervocalic "d" and final "t" from Latin intervocalic singleton "t" (realized as [ð] and [θ] respectively) were lost entirely, and [ts] was simplified to [s]. "Orderic and English" (Mark Faulkner) says the loss of [ð] and [θ] occurred "in the course of the twelfth century" (Orderic Vitalis: Life, Works and Interpretations, p. 118).
Also, the case system of Old French collapsed; most modern French words descend from the object case of Old French words, but a few show the form of the subject case. Ultimately, I don't think it makes a difference in this case because word-final [s] also ended up being lost later on, but this seems the basis of the modern French spelling of the (now phonetically identical) singular and plural forms of these words at any rate. At some point after [θ] and [ð] were lost, it seems that stressed "e" in an open syllable (so, word-final or followed immediately by word-final schwa) came to be realized automatically as high-mid [e] rather than as low-mid [ɛ]. In closed syllables (followed by a word-final consonant, or by a consonant before word-final schwa), raised "a" became [ɛ] or in some cases [jɛ] (Dell and Selkir have examples, but two off the top of my head are "aime" and "chien").
-tātem, -tātēs = -té, -tés
-ātum, -ātōs = -é, -és
-āta, -ātās = -ée, -ées
