Addition. Let me try to address Thierry Zell's concern about thethird question. To my way of thinking, the phenomenon ofthe question is an instance of the problem of uniformityof algorithms, a pervasively considered issue incomputability theory.
To illustrate, consider the question of whether a givenprogram $p$ halts on input $0$ before another program $q$.Let $f_p(q)=1$ if it does and otherwise $f_p(q)=0$. Everysuch function $f_p$ is computable, for similar reasons tomy $\pi$ function $f$ above, since either $p$ doesn't haltat all on input $0$, in which case $f_p$ is identically$0$, or $p$ does halt in $N$ steps, in which case we needonly run $q$ for $N$ steps to see if it halts, and give ouroutput for $f_p(q)$ by that time. So each $f_p$ is acomputable function. But the joint function$f(p,q)=f_p(q)$, a binary function, is not computable (ifit were, then we could solve the halting problem: to decideif $p$ halts on input $0$, design a program $q$ that wouldtake one step extra after a halt, and ask if $p$ haltsbefore $q$).
In other words, the function $f(p,q)$ is computable for anyfixed $p$, but not uniformly in $p$. And such uniformityissues are ubiquitous in computability theory.
In the example of the question, each class of graphs isdecidable, but not uniformly so, since by Tony's answerthere is no uniform algorithm, given a description of theclass, to find the collection of excluded minors. But forany such fixed class, the membership question is decidable.
The issue of whether a given algorithm is uniform in agiven parameter is a very common concern in computabilitytheory, and occurs throughout the subject.
