Do I have to write down contract rules on paper to ever get a real chance to fulfil them?

To get contract bonuses, one need to meet certain set of conditions. Is there any way to make a game remind / notify me, like:

You are at proper altitude and speed to fulfil "Test Mk16 Parachute in flight over Kerbin" contract.

Writing current contracts in real life notepad is really inconvenient, and checking against it is slow.

Nope. You don't need to write them down. In the top right corner, second from the right, you can click on that icon to give you a summary of your current contracts. As you fulfill conditions, the little clock icon will turn into a green checkmark. If you fulfill the root objective (ie. Test a certain rocket, deploy a certain part) once all the sub-objectives have green checkmarks, that will mark the contract complete.