No, only those items that specifically state they help pets will boost pet damage, etc.
Have a look at some of the various pet builds down in the Classes section of the forum as there are quite a few items listed which will help pets. JayNyne’s Lazy Pokemon build for example which I’m using for my own Summoner toon. Also look at the Devotions given in builds as these will also help pets.
Look for items with these as either affixes or suffixes: Beastcaller’s, Subjugator’s … of the Wild. Also the Rhowari faction has good items for pets as does the Black Legion so get your reputation with them high enough to buy these.
And no again, all element damage on gear isn’t useless as there are other builds with or without pets that would use these items.
Pet items are a little bit more difficult to find, but also check the vendors as they sometimes have items too. Bloodsworn Codexes/Tomes which fall from Cultists can also have pet bonuses on them. “of Bestial Rage” is another suffix to look out for.