Way to know the total development of a specific country

There is no auto explore button as there is with protecting trade etc.

However you can define multiple destinations by holding down the shift key whilst plotting your course. This allows you to select multiple ocean tiles. Therefore you can customise the course you wish a ship to take over the coming weeks/months by selecting lots of different destinations in one go.

The same applies with land units.

For Naval units there is also the Patrol option. Which once selected will cause the ships to repeat their orders in a loop. This can include docking at a friendly port for repairs.

A combination of using the Shift Key to stack orders and plot your desired course and then using the Patrol option to repeat those orders can be very handy and make Naval operations such as exploring a doddle rather than a chore.

I use this a lot when exploring, by making a ship sail up and down the same stretch of coast line over and over again. Then leaving him to it. You've just got to mindful of the ships attrition and make sure it visits port for repairs regularly enough.

Update - Limited to El Dorado DLC only

As @Max stated, you can now order fleets with explorers to automatically explore uncharted territories and coasts without taking attrition.

"Fleets with an explorer and at least 3 light or heavy ships (or a mix of both), that are in a port can undertake exploration missions such as exploring a sea, charting a coastline and circumnavigating the globe. When on an exploration mission, they do not suffer from attrition and cannot divert from their course."

See the Naval Exploration section of the EU4 wiki

The disasters are well documented in the EU4 wiki.

To take your peasants war as an example, here's how it should work:

Pre-requisites
- Is a monarchy (except Celestial Empire and Steppe Horde).
- Has greater than 5 provinces.

Progress
- Has no disaster ongoing.
- Manpower less than 25.0%.

Monthly Progress
0.5% when: not at war, overextension between 25% and 50%
1.0% when: not at war, overextension more than 50%
1.0% when: not at war, stability less than +1
1.0% when: not at war, war exhaustion more than 10.0
1.0% when: not at war, has taken more than 10 loans
1.5% when: not at war, legitimacy less than 50
2.0% when: not at war, nation is bankrupt

Effects
- +5.00 Unrest
- +50% Stability cost modifier
- Peasant's War random events.

Ends when
- No revolts are active.
- Stability at least +1.

So, the disaster should go away, if your manpower exceeds 25%. If it has already fired, you need to first defeat all revolts, and have a stability of +1.

Source: http://www.eu4wiki.com/Disasters