What are various pros and cons of granting or rejecting military access?

Solution 1:

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.

Solution 2:

As per the article on diplomacy on the EU4 wiki:

Allows you or the target country to pass through each others territories. Reputation is increased by 10 until the military access is revoked by either country.

The pros are a gain of reputation with the nation you're allowing military access with, this will be forfeited if you cancel the agreement. Additionally, as mentioned in the article on relations (also on the EU4 wiki):

If you have military access agreements, you can gain trust by beating rebels in their provinces or winning sieges in their provinces