purse, hearse, terse, worse, first. Are there other vowels we write for the same sound before rs? Why is there such a variety?

Indeed, those pesky spellings of the vowel sound /ɜː/, so well illustrated by the sentence: "Shirley, the perfect nurse, turned into the worst person in the world and murdered a third of the personnel at work." Now for your questions:

1. Are there other vowels we write for the same sound before rs?

Yes, apart from the -ea-, -e-, -i-, -o- and -u- you have mentioned, there's the "y" in "thyrse", a botanical term meaning "a type of inflorescence; a compact panicle having an obscured main axis and cymose subaxes." See: http://en.wiktionary.org/wiki/thyrse

The "y" in "myrrh", "myrtle", "Myrmidon" and "thyrsus" also produces the sound /ɜː/.

2. Why is there such a variety?

Two possible reasons. The first is that words such as "purse, hearse, terse, worse, first" might have been pronounced differently when the spelling system of English was first standardised.

The second is that English spelling tends to reflect the orthography of the language from which the lexical item was derived, complicated sometimes by "folk etymologies", where ordinary people assume a common origin for words that are really unconnected and spell them accordingly. Here are the etymologies of the words you have listed:

Purse: From Middle English, from Old English purs (“purse”), partly from Old English pusa (“wallet, bag, scrip”), and partly from Old English burse (“pouch, bag”).

Hearse: From Old French herce, from Medieval Latin hercia, from Latin herpicem, hirpex

Terse: from French ters (“clean”), from Latin tersus (“cleansed; neat, spruce”)

Worse: From Old English wyrsa, from Proto-Germanic *wirsizô. Cognate with Dutch wers (“worse”)

First: From Middle English first, furst, ferst, fyrst, from Old English fyrst, fyrest (“first, foremost, principal, chief, original”), from Proto-Germanic *furistaz (“foremost, first”), superlative of Proto-Germanic *fur, *fura, *furi (“before”), from Proto-Indo-European *per-, *pero- (“forward, beyond, around”), equivalent to fore +‎ -est. Cognate with North Frisian foarste (“first”), Dutch voorste (“foremost, first”), German Fürst (“chief, prince”, literally “first (born)”), Swedish första (“first”), Icelandic fyrstur (“first”)

Spencer Tracy said of Katherine Hepburn; There's not much meat on her, but what there is is cherce.