Ability to check where my site ranks on main SEs for a specific search term.
Yes, there are hundreds of these tools available. Search for 'keyword ranking tool' that type of thing. I would suggest using one that is search engine friendly, they're available.
- Ability to check what keywords are generating more traffic to my competitors.
- Ability to check what sites are generating more traffic to my competitors.
Hmmm...not sure if I'm understanding your question, but if I do, the only way to learn this information would be from your competitor's search logs. And I doubt they will share that information with you;-)
The best of all, would be a tool that would test scenarios for changes on my SEO strategy. Let's say I used to say BLUE WIDGET and want to change DISCOUNT WIDGET, then program would show me the expected trend.
I've never heard of this type of tool, although if it exists I can see how it would be helpful.