If you buy an external DAC, it will have to have a headphone amp built in to be able to play headphones, or you need a separate external headphone amp. I think Topping might have t-amps with headphone amps built in (not sure) but would probably take all of your budget.
Not sure what you mean. If you buy passive bookshelf speakers, you have to buy an amp to power them. Otherwise, you have to buy powered speakers. Given that you have only a $100 budget, I don't see how you could buy a DAC with optical inputs and headphone amp, and a t-amp, and passive speakers.