In a Zener card test using a 25 -card pack and no replacement, what is the expected score if we try to minimise correct guesses?

As with the maximization problem, this admits to a recursive solution through what is effectively dynamic programming. Let the number of remaining cards of each type be ${\bf n}=(n_1, n_2, n_3, n_4, n_5)$. The probability of type $i$ coming up next is $p_i({\bf n})=n_i / \sum_j n_j$; if type $i$ comes up next, there will be ${\bf n}-{\bf e}_i$ remaining cards; and if the guess was $j$, then the change in score is $\delta_{i,j}$. So we can say that the expected score with best play satisfies $$ S({\bf n})=\min_j\left(\sum_i p_i({\bf n})\left[\delta_{i,j}+S({\bf n}-{\bf e}_i)\right]\right)=\min_j\left(p_j({\bf n}) + \sum_i p_i({\bf n})S({\bf n}-{\bf e}_i)\right) $$ if you're trying to minimize your score, and $$ T({\bf n})=\max_j\left(p_j({\bf n}) + \sum_i p_i({\bf n})T({\bf n}-{\bf e}_i)\right) $$ if you're trying to maximize it. Python code to compute the result is as follows:

def p(ns, j): return Fraction(ns[j], sum(ns))

def dS(ns):
    nn = list(ns)
    ret = Fraction(0, 1)
    for i in xrange(len(ns)):
        if ns[i] > 0:
            nn[i] -= 1
            ret += p(ns, i) * S(tuple(nn))
            nn[i] += 1
    return ret

def S(ns, cache = {}):
    if sum(ns) == 0: return 0
    if not cache.has_key(ns):
        cache[ns] = min([p(ns, j) + dS(ns) for j in xrange(len(ns))])
    return cache[ns]

Switching 'min' to 'max' yields $T$ instead. We find that $S((5,5,5,5,5))=\frac{2048941091}{892371480}\approx 2.296063,$ and $T((5,5,5,5,5))=\frac{23148348523}{2677114440}$ $\approx 8.646753.$

Partial answer: the answer may be around 2.30.

I do not know how to show that rigorously; that was the figure I got from a Monte Carlo experiment with a million runs. I used the following guessing technique: always guess that the next card will be of the least numerous type remaining; or if two or more types are equally least numerous, choose the type among them that is leftmost on the list (square, circle, star, cross, wavy). The largest score was 5, the smallest 0.

2.30 intuitively seems a very low expected score!

Here is an example of a pack that gives a score of 0:

(cross, wavy, wavy, cross, star, star, cross, star, square, square, square, star, wavy, cross, circle, circle, circle, circle, square, cross, wavy, wavy, square, star, circle)