Although -- if the crust were entirely molten, that would make it too hot to carve through. Since (contrary to popular belief) vacuum is a superb insulator, a molten asteroid would take a long time to cool down and resolidify, especially if it were drawing closer to the Sun. So the idea might be that it would remain molten for however long it took to spiral down into the Sun.
1. We never saw the projected time for it to fall into the sun based on their plan.
2. We don't know how fast the technology in the setting is at tunneling through rock.
3. I would guess the characters had some idea their plan would work.
Both of your comments get to the capabilities of this future, fictional technology, which leaves some leeway about the feasibility of this plan. A couple of thoughts suggest it's not that feasible.
In terms of the bombs, we don't know their yield, but there are realistic constraints as to how powerful they could be. They really can't be more powerful than nuclear bombs of that size. Nukes don't create gobs of lava, so I don't think these would either. If there were magma, thermal radiation (rather than conduction or convection) would be the primary means for it to cool down via the vacuum. I don't know how long that would take but it would depend on the amount of magma. I don't think there'd be much and probably the molten material would be blown away by the force of the explosion anyway. (Although, after I think about it, vacuum isn't a good heat conductor, but the extremely cold interior of Eros probably is. It might cool down any magma fairly quickly. )
In terms of imparting a significant delta vee in Eros towards the sun, it's hard to say again with the fictional technology. However, it's hard to imagine imparting a large amount of velocity to such a massive object (~14,800,000,000,000,000 pounds) using any sort of reality based technology. They can redirect a little but probably not accelerate it much. Eros orbits at about 54,000 miles/hour, so it'll take a long time to go the 135 million miles. Of course, it would gain speed as it fell towards the sun. However, I have a hard time picturing them moving Eros perpendicular to its own orbit to go directly into the sun.
At any rate, for the sake of the story, I'm willing to buy the fact that Fred Johnson's geeks crunched the numbers and found that it would work--if Eros hadn't sidestepped the ship!