Granting and revoking military access is mostly a strategically important tool. The reputation hit or bonus are usually too minor to be of much consequence. It's best to check the reasons for requiring military access before granting or refusing it to support the player's best interests. Generally, it's best to grant access to parties in wars outside the sphere of interests of the country or to participants in a war against a major rival (the more they fight, the less powerful they become), and refuse access to aggressive rivals and hostile nations (they hate you anyway, and you want them to be weak).
For example, imagine the following: Country A borders country B, and country B has exclaves inside the territory of A, while A has cores on those very same provinces. Rebellion starts in the exclaves, with the intent to defect to country A. If A grants military access to B, they can crush the rebellion and keep the territory. However, if A refuses, there's no way B can crush the revolt, even if it's very minor, and the provinces will eventually defect to A without war. Refusing land access can also be used to protect minor states that one has appetites in subduing, but are also the target of other countries. In this way, even minors can chip away territory from the major players, if there's a hostile nation sitting in between.
Another use case is to support strategic partners or prevent rivals from participating effectively in a war you are not involved in. Land access is often critical to the success of a campaign, and AI rarely builds massive transport fleets to haul significant number of troops around.
The issues of military access show that it's vital to keep the realm connected by land and optionally by sea without the assistance of a third party. A good example of this situation is Burgundy in 1444: the Burgundy region and the Low Lands are separated, with HRE minors in between. If Burgundy angers them for some reason (but is not at war with them), a third party (e.g. Austria) can attack Burgundy region and lay siege to it without a chance for troops stationed in the Low Countries to participate.
First, a small correction: Protestant faith has 10% reduced idea cost as opposed to technology cost. This is still pretty powerful bonus, as there are very few other ways to reduce the idea cost, and it's typically not very difficult to keep abreast with technological developments if you are in a Christian-following technology group.
Let's list the properties of each denomination of Catholicism first:
Catholicism
- Tolerance of the True Faith: +1 (i.e. -1 revolt risk in the province of your own faith)
- Tolerance of Heretics: -1 (i.e. +1 revolt risk in provinces of other Catholic denominations)
Apart from these bonuses, you also have a possibility in participating in Papal politics -- controlling cardinals and ultimately the Curia itself.
Controlling cardinals confer small, but appreciable bonuses to prestige, legitimacy, papal influence and even technology cost. The Papal controller also gets an additional diplomat, larger advisor pool, additional leader slot, more papal influence and prestige. On top of that, the Papal controller can dissolve royal marriages without a stability hit, can excommunicate catholic rulers and can call crusades (until 1650). Pretty significant laundry list of bonuses, to be sure -- only if you play and win the Papal game.
Protestant
- National Tax Modifier: +10% (a moderately powerful boost)
- Idea costs: -10%
Switching the state religion confers also a significant lump sum of cash, as well as a heavy stability hit. Apart from that, there's not much additional benefits for protestant nations.
Reformed
- Trade Efficiency: +10% (interesting choice only if you are trade-oriented)
- Advisor Pool: +1 (one additional advisor to choose from)
Similar to Protestant, you receive a substantial amount of money upon switching, together with reduced stability.
If you are a minor nation, it's going to be difficult to have more than one or at most two cardinals in the Curia at any given time, and becoming the Papal controller is probably a pipe dream. So, if you can't compete with the big boys and everybody else around has converted, it might be a good idea for you to do that too.
Countries in the Germanic, Scandinavian, British and West Slavic culture groups are much more likely to switch religion than others; conversely, Latin and Iberian cultures are much less likely to do so. If you play Bohemia, for example, it will be virtually impossible not to switch to Protestantism.
If you are the Catholic Defender of Faith and/or control the Curia, spontaneous conversion of provinces cannot occur[verification needed].
Other minor factors also control how often provinces switch, like whether you have or not Religious or Innovativeness ideas.
If you are determined to stay Catholic but a lot of your provinces start to switch, wait for the Counter-Reformation event, which fires around 20 years after the start of the Reformation. If you take the decision, you will receive additional missionaries and increased missionary strength, at the cost of slightly increased future technology expenditures.
Be advised that switching religions is going to be a tremulous time for your realm: expect events reducing stability and increased revolt activity for a number of years. Don't take the decision unless a sizeable portion of your nation has flipped or if you are in a precarious position (overextended, targeted by a coalition, depleted manpower pool).
TLDR;
- If you have Germanic, Scandinavian, West Slavic or British culture and are not powerful enough to consistently control the Curia, it's probably best to switch.
- If you are trade-oriented or a republic, switching might be best for you too.
- If you are a pious Catholic from France, Iberia or Italy, the Defender of Faith or control the Curia, stay catholic.
- If you are determined to stay Catholic in the face of significant province switching, consider embracing the Counter-Reformation.
Best Answer
Here's what I'd recommend. First, a small country is always good, because you can have a pretty strong hold on them. One province minors are particularly useful. I would recommend you have an alliance and royal marriage with them, because, aside from the relations bonus, alliances give you a +50 score and royal marriages give you a +10 score. Also, you should probably guarantee their independence in order to prevent them from being conquered. Also related to that, try to find a nation that is within the middle of the HRE, nations on the borders, especially the border with France, are much more likely to be picked off. The last thing that can help to make them loyal to you is to make sure they're part of the same culture group as you, which gives a +5 bonus.