Can't you just point it there, fire a thruster, and let it go it's course? (I mean, calculate where it should end up, but yeah, basically go towards the sun.)
Nope. I think Larry Niven covered this too, prolific guy. You either speed it up, which takes it to a higher orbit or slow it down which takes it to a lower orbit. To take it to an orbit low enough to burn up in the sun, you need to slow it down the whole amount that the earth is orbiting the sun (about 67000 mph). There's a little bit of fudge in this due to eliptical orbits and drag from the sun's atmosphere but that's the basic principle. Essentially anything you fire into the sun is going to get there on a spiral path.
Correct. I really recommend reading up on the Messenger probe that went to Mercury, it needed an insane amount of fuel and a number of earth and venus flybys to slow down enough to even get to mercury. In space speeding up takes energy, slowing down takes energy, everything takes energy :-)
In terms of space stuff remember that we lived to see the Viking landers. Those were pretty darn amazing.
In terms of great filters and the Fermi paradox, remember that one reason we might not see others is that they're just more efficient. Take lighting: For a long time the best way to see life on earth is to do a spectrographic analysis of our planet and you would see an insane line for sodium which is not natural. That's the sodium vapor lights we used for street lamps. Except now we are using LEDs which have far less light scattered upwards because they are much more efficient.
Or TV signals: Our TV signals used to be easy to spot: A massive signal spike (the sync signal) followed by colorburst, then the X and Y of the scan line. Easy to spot, very repeating pattern, can't be nature made, obvious sign of intelligence. However now we have switched over to HDTV which is not only digital but encrypted digital so the stuff looks like background noise. And it's much lower power because once again efficient.
That's in 100 years. There may be only a short window where we can spot other civs before they get efficient and encrypt everything because the Altarian Disney Company doesn't want to share their content without payment :-)