If your TV has stereo outputs with variable volume, so that you can
control the the volume through the TV and the remote, I'd go with
option #3 and some 2.1 PC speakers.

I've done that several times myself. Works very well. PC speakers are
small and are magnetically shielded, so they won't interfere with your
TV.

Option #2 would be my second choice. By shopping flea markets, pawn
shops, thrift stores, and eBay, it should be easy to find an older
integrated amp or receiver for $25 and up. (Keep in mind that with
eBay, you'll need to pay shipping.)

Option #2 will give you more options and flexibility, but will be
bigger and clumsier too. If all you want to do is watch TV and DVDs,
option #3 is the best way to go.

I know this because I've tried all of the above. I've used option #3
several times myself and it's the smallest and most convenient.
Convenience is important to me because I have a family and they want to
be able to turn on the TV and listen and watch, not fuss with a lot of
audio gear.

Altec, Cambridge Soundworks, Klipsch, Logitech, Sony and others make
some good PC satellite/subwoofer speakers that should work well for
you. All are available cheap on eBay, but remember to check the
shipping cost and the reputation of the seller.

I'd ignore the watt ratings. Just buy whatever sounds good to you. The
sound quality will very likely be much better than what your TV can
provide.

Yet another thought: Because you have a PC, and it probably has a DVD
drive, you could just add a TV tuner to your PC and use the PC for TV
and DVDs too, using PC speakers. (I also do this.) However, the TV
image on your PC monitor probably won't be nearly as good as the image
on even a cheap TV, although DVDs should look good. Also, watching TV
and DVDs on a PC can be pretty clumsy.