P&P is, indeed, primarily a 'projectionist frame', in your terms, to the extent that in P&PT syntactic construction starts with, and proceeds 'bottom up' from, a choice of lexical items, by gradually 'satisfying' their complementation/selection features. (This applies to adjuncts, too). To the extent that functional heads like Aspect(s), Modaliti(es), Polarity, Tense, C, etc. are also considered lexical items (which is legitimate, since they ultimately reduce to sets of valued, or to-be-valued-during-a-successful-derivation, attribute-value pairs), P&P is an almost 'pure' projectionist model. The only caveat with respect to such a categorization follows from the fact that the order in which some functional heads are 'projected' relative to each other does not yet follow from their lexical properties (plus general principles like 'satisfaction') and, as a consequence, it has to be stipulated by means of ad hoc 'templates' of the form 'C1 must c-command/have scope over C2', .... 'Cn must c-command/have scope over Cm', etc. Such 'templates', of course, are construction-like and, to the extent P&P syntax would not at present work without them, P&P remains 'constructionist' to a certain degree (like Tagmemics, other 'structuralist' 'slot and filler grammars' of the 1950' and 1960's, or, in fact, early TGG), but that is so because P&P has not yet entirely succeeded in making the whole hierarchy of projections follow from lexical properties and general principles. Programmatically speaking, P&P aimed at becoming a pure 'projectionist' model of syntax.
As to your second question, obviously, '(neo)constructionist' approaches, like Tagmemics or structuralist 'slot and filler grammars' of the last century, reject the claim that syntax follows from the projection of lexical properties, rely on 'top-down' or mixed (up&down/parallelistic) generation strategies, and, on the contrary, have themselves largely emerged as a reaction against GB and P&P models of syntax, not the converse.
Finally, the 'minimalist program' is by no means a reaction against 'neoconstructionism'. If anything, the MP can be described, for current purposes, as a reaction against previous 'projectionist' Chomskian work along GB and P&P lines. Of course, under minimalist assumptions, syntactic derivations still start from a selection of lexical items (a 'numeration', in the earliest minimalist work, a 'lexical array' in later versions, or from parallel access to the Lexicon in some alternative versions), but Minimalist Syntax, especially in its mature formulations, is no longer 'projectionist' (at least not in its programmatic statements) because the only syntactic operation it allows is (internal/external) 'Merge', and, in theory, 'Merge' is just a recursive set-forming operation subject to 'No Tampering', a principle that claims that the internal features of lexical items cannot be modified by Merge in any way. Thus, in theory at least, minimalist syntactic construction is no longer a matter of 'projecting' and 'satisfying' the features of lexical items, be they 'descriptive' or 'functional'.
Whether MG delivers what it aims at is a different matter, though. I wrote 'in theory' and 'programmatically' above because even MP cannot dispense with 'labels' (well, an attempt was made to get by without them a few years ago, but it was a sort of trick and did not catch on, to my knowledge), and the 'label' attached to a merger is 'inherited' from one of the syntactic objects (at first lexical, subsequently phrasal) that Merge operates on, which entails that at least that feature of one of the two objects merged must be searched for inside the object and 'copied'. It is true that, according to Chomsky,'copying' F is not 'tampering' with F, or with the object that contains F, since the original object is not really modified, just duplicated. However, in cases of internal Merge, where the'checking' of a valued feature [F:v] or the 'valuation' of an unvalued one [F:?] is involved, the 'No Tampering' principle is untenable, in my view, because in such circumstances the internal structure of the corresponding categories is inevitably modified by the checking/valuation process (which is why, once F is 'checked'or 'valued', it is 'deactivated' and cannot be 'checked/valued' again - a crucial condition for the correct behaviour of movement, agreement and other 'operations' currently considered aspects of Merge).
In sum: in spite of the official programmatic statements to the contrary, even minimalist grammar remains 'projectionist' to some extent. What MG is not at all is a 'constructionist' model, and, needless to say, it did not arise as a reaction against 'neoconstructionist' approaches.