Re: 6: You're right, Friedman certainly has something to say on this topic! Friedman has discovered a number of $\Pi^0_1$ sentences of Ramseyish flavor which have large cardinal strength. For example, say that a function $f: \omega^k\rightarrow \omega$ is good if there are rational $p, q>1$ such that $$p\vert \overline{x}\vert<f(\overline{x})<q\vert\overline{x}\vert$$ (where $\vert \overline{x}\vert$ is the sup of the elements of $\overline{x}$). Then Friedman showed that the following statement is equivalent (over $RCA_0$, I believe) to "For all $n$ there is an $n$-Mahlo": $$\text{$\forall$ good $f, g: \omega^k\rightarrow\omega$, $\exists$ infinite $A, B, C$ such that $f[A]\subseteq C\sqcup g[B]$ and $f[B]\subseteq C\sqcup g[C].$}$$ (Note that folded into the conclusion are the statements that $C\cap g[B]=C\cap g[C]=\emptyset$.) In general, look at his book "Boolean Relation Theory" (available on his website), or - maybe more digestible - some of his numerous postings to FOM (http://www.cs.nyu.edu/pipermail/fom/); I don't know how high up the large cardinal hierarchy he's gotten, but I believe it's quite a ways.
Now an aside: this certainly isn't finite or discrete math, but you might still find it relevant. There are a number of occurences of large cardinals in questions around left distributive algebras. For example, see http://www.ams.org/journals/era/1997-03-04/S1079-6762-97-00020-6/S1079-6762-97-00020-6.pdf by Dougherty and Jech, which shows that the freeness of a certain left distributive algebra $A_\infty$ follows from a large cardinal assumption, and is unprovable in $PRA$ (quite a gap!); I believe the question of whether the large cardinal assumption is necessary is still open. More famously, the word problem for the free left distributive algebra on one generator was originally solved by Laver using large cardinals; it was then given an elementary proof by Dehornoy.
Re: 1, an interesting cautionary tale: if I recall correctly (I don't have it on me at the moment), in his thesis Reinhardt gives an argument - which I at least find pretty nice and compelling - for why we should accept large cardinals arising from better and better elementary embeddings of $V$ into subclasses of itself. This concludes with an argument for the ultimate large cardinal of this form, a $\kappa$ which is the critical point of some nontrivial $j: V\prec V$. Of course, $ZFC$ disproves these cardinals, so there is an example of a nice philosophical argument which ultimately runs off a cliff.
