Oh dear, rain barrels will not work with regular sprinklers. But if you place them higher than the ground. Preferably 10" or more than you can easily fill your watering can. It will work with drip or soaker line without the pressure regulator but you will need to install a 150 mesh or 200 mesh filter.

And as you found out, you do need a screen filter at the inlet to the rain barrel. Also do check for mosquito larvae growing in them from time to time. A thin layer of cooking oil will usually take care of them.

before the water goes into the rain barrel
do need something like this http://www.rainharvest.com/filtratio...ut-filters.asp

when you want the water from the rain barrel to water your garden
do need something like this http://www.dripworks.com/category/filters
don't need http://www.dripworks.com/product/323...ure-regulators

And no you can't connect into your existing irrigation system without check valves and few other things.