The Wikipedia article about regular languages mentions that $DSPACE(O(1))$ is equal to $REG$. Can I conclude from this that every function in $R$ with constant space complexity is in $REG$?
A regular automaton can do anything a Turing machine can do, as long as the TM uses only O(1) memory. This is because with finite memory, the number of possible states the TM can be in is also finite (It's the number of possible tape contents * the number of possible head positions * the number of states of your TM). You can encode the whole computation graph as one big finite automaton.