As @GMJoe mentions, by choosing a power with a domain on the Outlands you are only one plane removed from all of the planes on the outer ring.
Because powers are (or can be) actively involved on the planes, they are a great trigger for adventures (even if the party does not have a cleric). A cleric has to have a good reason to refuse a request from his power to get involved in something. In our case this usually also means we have support from the power (+pantheon) in our adventure. This can be in magic items (also see next point), in assistance in getting around the planes, or whatever.
Powers can give their followers Power Keys which enables clerics to cast without penalties on other planes (see page 14 in DM guide of the Planescape setting) or even to enhance certain spells (e.g. always maximum effect). These things should be rare and their use should be limited because they are very powerful. You could think of giving your cleric a Power Key for the duration of a mission undertaken by the party for the Power; or perhaps tie it with a specific plane (i.e. it only works on Baator). Great opportunities for story.
Also, it are not just the clerics whose magic are affected. Other magic is also affected by the planes (see table II in the DM guide of the Planescape setting). For example, illusions don’t work on Mechanus. This effect is, however, not as dramatic as the effect on clerics can be in some planes.
The effects of planes on magic and armour are not too difficult to keep track of. The armour class and attack roll are for the plane of origin of the armour/weapon and I subtract the appropriate penalties from my roll.
Besides magical weapons and armour, other items are also affected just like magic performed by wizards and sorcerers (as mentioned earlier). From the DM guide: ‘A wand of wonder, which uses wild magic, is a useless stick on orderly Mechanus. And unlike spells, there are no keys to make magical items work properly.’ So it is actually a bit more complex, but also fun.
Your other points are a bit difficult to answer without concrete question. For the ‘Power of Belief’ I would like to mention that a classic adventure in Planescape is preventing or causing a certain location to planeshift.