In this case,de quoi is a relative substantive clause (relative substantive in French or relative sans antécédent). It complements the verb donner in the sentence

Il lui donnait juste de quoi vivre.

According to Grammaire méthodique du français by Martin Riegel, Jean-Christophe Pellat and René Rioul, relative substantive clauses are introduced by a pronoun, here quoi, which has to be preceded by a preposition, in this case de, since the relative pronoun is a non-animated object (money, a flat, food for example). Moreover, the tense verb of the relative clause vivre, is infinitif, since donner is a transitive verb. The tense has to be infinitif if the relative is preceded by il y a, such as in

Il y a de quoi manger.
Il y a de quoi rire.

About the first example you mention,

As-tu de quoi écrire ?
avec is omitted, because it is obvious the object denoted by the relative pronoun is to be used to write with (paper, ink, pen…).

Regarding your second comment (the disappearance of one de), it is known in French as la règle de cacophonie, namely, that repeating de/des would not be very pleasant to hear or utter, mentionned in la Grammaire générale et raisonnée de Port-Royal :

de des, et encore plus de de, eût trop choqué l’oreille, et elle eût peine à souffrir qu’on eût dit : il est accusé de des crimes horribles, ou, il est accusé de de grands crimes.

avoir à faire is indeed the correct way to say to have something to do. We would less laconically say avoir quelque chose à faire to mean the same thing.

Il y a de quoi faire, j’ai de quoi faire is rather a casual expression, with an underlying nuance of awe, admiration, annoyance… See for example http://www.languefrancaise.net/Bob/37467.

For example, we could say in front of a gargantuesque quantity of food during le réveillon de Noël:

eh beh, il y a de quoi faire !

implying, wow, such a great quantity of good-smelling food!
Or when having to clean the mess after the dog wreaked havoc in the living room:

Purée, il y a de quoi faire !

or before undertaking a long, straining task,

Il y a de quoi faire !

Note that it differs from expressions such as:

J’ai de quoi faire parler cet homme

for example, meaning in this case I have means to make this man talk.

Il lui donnait juste de quoi vivre.

Literally: He him gave just (1) of what to live.

(1) Understand “just enough”.