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Generally speaking it's not that big of a factor as generally the constant cycle of battles (and potentially colonization) will keep your prestige pretty high if you win most of them.
As for diplo power, you're main ways of increasing diplo power are:
1. Pray for a better ruler
2. Shell out more money for a better diplomatic advisor.
3. Make it your national focus (but this also reduces your administrative and military power).
The wiki is extremely useful for most information, though it gets outdated each new patch/expansion.
http://www.eu4wiki.com/Europa_Universalis_4_Wiki
There are several modifiers to it, but by default you can only have relations with four different nations, for every one above that you have to pay one diplo point per month.
It's similar for military points and leaders.
http://www.eu4wiki.com/Income#Income
Quick ways to increase your income is to reduce military spending. This reduces the available morale of your units. But is usually a very good idea during periods of peace as it frees up alot of income. Note that at 0% military funding your troops wont reinforce and wont be able to move.
for full info on maintance effects look up
http://www.eu4wiki.com/Army#Maintenance
Addendum:
A sneaky way to hasten the death of a ruler or an heir is by making them into a military leader(general) in the military tab. Ideally putting them in command of a unit and then just let them sit there. The chance of a military leader dying is alot higher then that of a regular ruler or heir.
When a colonizing nation gets 5 cores in one colonial region a colonial nation will form as a subject nation. If they get pressed for money too much they may declare independence. Depending on the starting date there may also be colonial nations already starting independent.
Without CoP you can only play a colonial nation that's independent, CoP allows you to play one that's still a subject nation.