Yes. An opponent's Obelisk or Ra can be tributed as cost for your Lava Golem. It is not an effect, nor does it target, which is why Obelisk is not protected from it. Ra has no protections anyway apart from during his summon.
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A 'tribute' must be something you control. You can only tribute something you do not control, if the card specifically says you can, such as Lava Golem.
You can never tribute an opponent's monster unless the card specifically says you can (Lava Golem, Soul Exchange).
You can never tribute something you don't control unless the effect specifically lets you.So Soul Exchange, Trap Eater, Lava Golem are all examples of something that allow you to tribute your opponent's cards, and they all clearly say so.But if there is no text allowing it, then by default, it can only be a card you control, ie, something on your side of the field.
Unless stated otherwise (Plaguespreader Zombie for example) a monster's special summon is done from hand only. Lava Golem must be in your hand, for you to be able to special summon it using the listed text on the card.
In either of your own main phases, you can special summon lava golem. Rather than offering the cost from your own side of the field, Lava Golem says to use the opponent's monsters. So you send the opponent's two monsters to the graveyard and place Lava Golem on their side of the field. Either player can try to negate it, in which case it will be destroyed and go to your graveyard.