The 10% or 20% refers to the culture
slider. This is a slider that appears in the top left of the screen (next to the research slider) after you research
Drama. When you increase this slider above 0%, you start claiming the happiness bonuses from Colosseum and Theaters.
Generally it's best not to use the culture slider too much unless you're (a) suffering from a lot of temporary unhappiness e.g. war weariness, or (b) heading towards a cultural victory. Otherwise your research rate will suffer.
The option for Steal Tech is only available if there is a tech which that civ has that you don't have,
and that you have all the prerequisites for. I think Influence Religion is only available if the two of you have different religions. Sabotage Improvement only works if you move your Spy onto a tile with an improvement you want to pillage. Sabotage Project only works if the player is building a Project in that city - which will only happen right near the end of the game, and mostly correlates to sabotaging the Spaceship (the parts of which make up most of the Projects in the game).