None of these can be explained by synchronic rules of Latin grammar, at least not in the classical age. Nor does this seem to be regular Proto-Indo-European ablaut, where one would expect something like **cp-/cep-/cop-*. Somehow many different allomorphs of caput exist; but allomorphs of other stems are plentiful in Latin and in most other languages. Many combinations of these allomorphs appear to be possible:
- cap-/cep-/cip- (semantic root)
- -ut-/-it- (suffix)
As to the nominative singular -ut v. stem -it-, nominatives singular are normally different from the other forms of words or suffixes of the 3rd declension. Homo, hominis; rex, regis; munus, muneris; etc. Each nominative can probably be explained, and there are patterns, but I don't know about the -u- in caput specifically. And -ud/-ut is probably a Proto-Indo-European suffix, probably with its own rules. I have consulted Walde-Hoffmann (1938) and Ernout-Meillet (1959), but I couldn't find a real explanation for -ut- v. -it- in caput, capitis.
Interestingly, the same -u/i- allomorphs seem to exist in a Germanic word that is probably cognate to caput (so says Philippa (2004–2009), although the Oxford English Dictionary (2nd ed.) is sceptical). Dutch hoofd and German Haupt come from Proto-Germanic **haub-id-*, whereas Old English heafod, modern head, and Old Norse haufuþ come from **haub-ud-*. According to the Oxford English Dictionary, this -id/ud- constitutes "suffix ablaut". One would expect the Germanic and Latin suffixes to be cognates too, but, alas, etymological dictionaries rarely explain all the affixes, and I couldn't find it. Note that there is also a Germanic allomorph **hab-* leading to Proto-Germanic **hab-ud-*, Old English hafud, and Old Norse höfuþ.
As to the -cip- in praecipito, an -a- in the first syllable of a verb normally changed into -i- after a prefix, in a certain phase of Latin. The same applies to -ae- and -e-, but not to -o- and -u-, as far as I know. The length of the vowel was generally preserved, so that long -ā- and -ae- become long -ī-, but short -ă- becomes short -ĭ-. Note that vowels did not (always) change in words formed in later phases of Latin. Nor did they change in the perfect stems of verbs. Hence:
Maneo - remaneo - immanens
Paro - reparo - apparans
(Present) capio - recipio - incipiens
(Perfect) cepi - recepi - receptus
(Present) ago - redigo - redigiens
(Perfect) egi - redegi - redactus
This may explain -cip- after many suffixes. However, it is possible that it is simply an older, unexplained allomorph.
As to -e- in praeceps, this appears to be an allomorph that already existed in prehistoric Latin or earlier. (Cep- in the perfect is probably a lengthening of the vowel and explained through other mechanisms, but this is not certain.) The dictionaries above have no real explanation. It is possible that the -e- is due to contamination from words like princeps and auceps, which probably came from the e grade of the root of capio. According to Ernout-Meillet, the root of capio had three allomorphs that seem to display regular Proto-Indo-European ablaut: **kēp-, kōp-, and **kəp-* (from which cap-). Notice that the stem of princeps etc. is -cip-, whereas that of praeceps etc. is -cipit-.
The -e- is only in the nominative singular, and it is not surprising in general that the nominative singular should be irregular. Pontifex (gen. pontificis) and maleficus come to mind, from facio, "make". Somehow both -e- and -i- occur. Compare also pars, partis leading to expers, expertis: the same nominative -e-, but no -i- in the stem.
As to the disappearance of the suffix in anceps, it is possible that it was syncopated, but it is more likely that the suffix was simply not used in all words derived from what may have been Proto-Indo-European **kh₂p*-/**kap-* (the origin of Proto-Indo-European **kap-ut* is not quite certain: **kh₂p*-/**kap-* may have come from a pre-Proto-Indo-European substrate language). This root meant something like "head", and later reflexes include some forms with the suffix -ut/-ud, others with -elo- (Greek kephalê), and probably others again with no suffix at all.
Note also that this root may be related to capio, habeo, and English/Germanic have/etc. (so suggests Walde), which come from a Proto-Indo-European root **kap-* (from **kh₂p-*/**kəp-*), or **gʰabʰ-* (from **gʰh₂bʰ-*) / **gʰebʰ-*, but this is uncertain. (These Proto-Indo-European roots may be variants on and reflexes of a pre-Proto-Indo-European substrate root, so suggests Philippa, Etymologisch Woordenboek van het Nederlands (2004–2009), 'hebben').
Words derived from the capio root exist that mean something like "recepticle, container" (notice cep-), and a semantic shift from pottery to head is a common phenomenon. Cf. Latin testa "(piece of) earthen pot", from torreo "burn" (related toast), which lead to French tête "head" and similar words in other Romance languages. Cf. also Latin cuppa "cup", probably from a non-Proto-Indo-European language (Philippa), which probably led to English cup, Dutch kop "cup; head (informal)", German Kopf "head (informal)". The original Proto-Indo-European root for "head" was **ker-*, "head, horn", from which horn, cerebrum, cervix, unicorn, triceratops; somehow the central word for "head" was replaced often in the successor languages.