1. the houthi are shiaum, that's pretty much it. the yemeni government has claimed it intercepted a ship that proves that iranian arms were being sent to the rebels, but they also won't give any information about the ship or present any other evidence. it really does look like they just made it up and that the entire basis for the charge is the fact that both the iranian government and the houthi are shia.
the problem is that the houthi are zaidis, members of a shia sect that is quite different from the twelver version practiced in iran. theologically, zaidi shi'ism is closer to sunni islam than it is to twelver shi'ism.
of course, the fact that they are different sects doesn't prove that iran isn't supporting them. there are plenty of examples of people from different religious sects cooperating for the sake of political expediency. it's just that the saudis and yemenis are not making much of a political expediency argument. they're just pointing to the fact that both groups fall under the category of "shia" and letting the implication flow from there.
on top of that, the saudis are really paranoid about iranian influence in the region these days. viewed through that lense, they're inclined to see almost everything as an iranian plot.
but now the iranians seem to have had enough. after being blamed for the rebellion for weeks, they've now unleashed their own rhetorical salvo and are charging the saudis with slaughtering shia. if they weren't already on the houthi's side, the saudis may have pushed them into it. all this time the saudis have been warning that the yemeni rebels could trigger a regional proxy war. it looks to me like the saudis may have created one themselves.
