What non-alphabetic characters are valid in English spelling?

No. A hyphen can appear in an English word as well. For example:

a five-year-old boy

There are the graphemes, œ and æ, which appeared first in medieval Latin to represent the Greek diphthong.

They are still used in everyday English (at least in the Queen's version), in words such as encyclopædia and fœtus. (At least they are usually written ae; those characters are not available on most keyboards).

I have noticed that Americans pronounce the first syllable of pædophile as one would pedestrian (suggesting pedophile - a lover of the feet?).

The reason they do this, presumably, is because they have lost contact with the diphthong.

See: http://en.wikipedia.org/wiki/Diphthong

And oh dear, Wikipedia has the same wretched shortcoming. Surely it should be Wikipædia, shouldn't it?

If we define "a correct spelling of an English word" as one that appears in a reliable dictionary, then, like ' (the apostrophe) and - (the hyphen), we must also include / (the slash) in the repetoire of valid English orthographic characters, because and/or appears in the Oxford English Dictionary:

and/or: a formula denoting that the items joined by it can be taken either together or as alternatives. Cf. either/or.

1855 Law Jrnl. Reports 24 ii. Excheq. 199/2 The parties were to ‘load a full and complete cargo of sugar, molasses, and/or other lawful produce’..the words ‘and’ and ‘or’ being introduced into the charter-party.

1895 F. Pollock & F. W. Maitland Hist. Eng. Law I. i. v. 152 In medieval Latin vel will often stand for and... Often it is like the and/or of our mercantile documents.

1916 H. Barber Aeroplane Speaks ii. 85 The jamming of the rudder and/or elevator.

1929 Penrose's Ann. XXXI. 99 A good proportion of cotton and/or linen in the furnish of a paper.

1941 Official Gaz. Kenya 13 May 305/2 Applicants are at liberty to submit their own proposals and/or programmes for the prospecting, development, and/or mining of the Owour Area.

1960 E. Bowen Time in Rome iii. 82 The young set-apart creature, waiting at home for her fifteenth birthday and/or the next vacancy in the Atrium.

1998 N. Lawson How to Eat (1999) 68 Grate in a cooking apple and/or a quince.