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A couple more remarks: (-) What A-M call modern may be old-fashioned now. Computer algebra and Groebner bases really took off after A-M's first edition. (-) Their statement is merely that primary decomposition is not 'such a central tool' anymore. One interpretation could be that it had a market share of 90% before and is now 50/50 with localization. That does not mean it's not important.
There is probably no debate that computing minimal primes (irreducible decomposition) is useful for applications: If a biologist comes with a bunch of polynomial equations and asks you to solve them, then you compute minimal primes. The crucial words in the A-M quote are probably "in the theory". I think their statement is too strong. In the beginning of Chapter 4 of "Commutative Algebra" Eisenbud calls primary decomposition the cousin of localization.
In a polynomial ring over a field the resolution has length at most $n$, that is $F_{i}=0$ for $i>n$. If $I$ is homogeneous, then there exists a unique minimal free resolution that simultaneously minimizes various measures of minimality. It turns out that this minimal resolution is a direct summand of any other resolution.
Hi variety, that's not a stupid question. The case only $ht(I) = ht(I')$ occurs when $I$ is already homogenous. I excluded that case in the beginning: I assumed that $p$ is non graded (and I'll also assume that $I$ is not graded, otherwise everything is trivial). We get $\text{ht}(IR_{(0)}=1$ since only the zero ideal has height zero and $IR_{(0)}$ is not the zero ideal since it contains a non-homogeneous element.
I think the question is too vague. Even in your new software you would need a module that does computer algebra and one that does bibliographies. How and why should they be linked? Everything except the semantic search is already there: The glue between the different "apps" is your operating system, for me Linux/text files/Emacs/shell. The semantic engine is tricky and part of ongoing research in computer science, but there are implementations like GnomeShell, MacOS Finder, and they work reasonably well if you keep all your stuff in text.
