"Secondary articulation" refers to vowel-like properties that are superimposed on consonants with various "primary" articulations. So "alveolar" or "bilabial" are examples of primary (consonantal) articulations, and "rounding", "palatalization", "pharyngealization" are secondary articulations. The best-known example of secondary articulation is palatalization in Slavic, where most consonants have plain versus palatalized versions.
In a number of languages, secondary articulations arise from assimilation to a neighboring vowel, for example in Russian, consonants are palatalized before front vowel suffixes such as the suffix for locative (prepositional) -e or abstract nouns -izm; or, in Nupe, consonants are rounded before round vowels (also palatalized before front vowels).
It is true that an assimilation is an effect of a preceding or following sound, which need not be immediately preceding or following. But almost all phonological processes involve a preceding or following segment, and many are not assimilations (for instance, dissimilation, not to mention ornery arbitrary changes such as in Chukchi, tʃ becomes s before q). Unless a rule is context-free (for example, g -> ŋ everywhere), any rule involves a preceding or following segment, so there isn’t anything special about assimilations or secondary articulations in that respect.
Given a two-way division of phonological properties into "quantity" and "quality" (basically, everything except length), assimilation does involve quality, insofar as quantity is a structural property that doesn't assimilate. It is correct that secondary articulation is about a specific type of place of articulation. However, secondary articulation and assimilation are incommensurable (one is not a case of the other). Assimilation refers to the process of one sound becoming more similar to another in any sense, and secondary articulation is a very specific fact about the articulation of consonants (whether via a rule, or simply lexically). Accordingly, Russian [bratʲ] 'to take' has a consonant with a secondary articulation ([tʲ]], and there is no assimilation (the consonant is underlyingly palatalized, and contrasts with [t]: [brat] 'brother').
The term 'coarticulation' is ambiguous between the specialized sense applied to [kp, gb] as in Igbo and other languages with labio-velar stop, where the term has been used at times to make it clear that the consonant in question is a single segment with two simultaneous articulations (not a cluster), and the more common contemporary usage deriving from phonetic theory, where the articulatory state of one segment perseveres or is anticipated, and overlaps that of another neighboring segment. An example of that is how the lips round in anticipation of a following rounded segment in English 'cool', 'school'. The process is continuous over time, as one can see looking at a spectrogram of 'school' and noting the downward-sloping line arising from increased labial constriction.
Coarticulation is similar to assimilation, except that assimilation is a categorial phonological change from one sound to another along a featural dimension, whereas coarticulation is a continuous change, and it is specifically in terms of articulators whereas features are only loosely related to particular articulators. It is hypothesized that coarticulation relates to assimilation, in that coarticulation is a natural phonetic phenomenon, and is often phonologized into assimilatory rules. Many assimilations are due to acoustic overlap, which, if phonologized, can cash out as a change in articulator state.
It would not be correct to maintain that coarticulation is physically mandated (this is essentially the SPE position). So-called vowel nasalization in English is clearly coarticulatory, as Cohn 1990 clasically shows, and it is specific to English (French has a different pattern). There are universally-mandated kinds of coarticulation, for example, in Turkish, there is lip protrusion during the production of consonants like /p,t,k/ when preceded and followed by the vowel [u] -- the lips do not suddenly un-round during the production of the /l/ in [pulun]. It is an open research question which instances of coarticulation are due to physical necessity, and which are part of the phonetics of the given language.
