Expenses
The top two things that cost you money in Civ 5 are:
To reduce the first one, be strategic about what you build where. Cities should be focused on a task. Pick terrain, tile improvements, and buildings that compliment this focus. For instance, if you are building a production-focused city, build it near hills and forests, and focus on military and production enhancing buildings. Don't build science or cashflow generating buildings, since you won't be producing much of those in this city.
Likewise, if you are building a science focused city, don't build military or other production enhancing buildings there. The city's focus should be on science or economic enhancing buildings instead.
The only exception is happiness - in order to maintain a positive balance on the happiness scale, you'll likely have to build some happiness generation buildings in most or all of your cities. This tends to be one of the largest trade offs economically in most of my games - do I make the people happier, or do I bank some extra gold?
You can also guide each city's governor to focus on a particular area, and this can help your citizens understand what they need to do in order to support your vision for the city. If you leave the governor alone, the city may ignore some of your tile improvements in order to work tiles that create something you're not interested in, but which makes the city's output more balanced.
Also, don't let your workers run free - focus your tile improvements on whatever works best for the city the tile is in the radius of. Automated workers don't always pick the best terrain improvement for you. They may destroy forest or plant farms around a city that is going to be production focused, for instance.
Every unit you produce costs you money every turn as well. Try to focus your attention towards a small, but mobile army, and don't fight on multiple fronts if you can avoid it. Mixed unit tactics involving ranged artillery units and close-up front line fighters work best. The AI is terrible at war, and they will generally start on the offensive, even though you've likely got your army massed and ready to funnel them into a meat grinder.
If you find yourself with a strong surplus of units, gift some to an allied city-state. You'll gain influence with them, and if you declare war, those units are likely to still benefit you indirectly.
Income
You gain money via citizens assigned to money-generating tiles, trade agreements with other Civs, and by trade routes between cities. Any duplicate of a luxury resource does you no good - it is there to trade.
If you end up with 2 of a luxury, or more of a strategic resource than you need, find someone rich and sell it to them for a tidy profit. Suppling your possible enemies with strategic resources might sound like a bad idea, but since you control the flow of that resource, if they start doing something you don't like, you can cut them off and drastically reduce the effectiveness of any resource-requiring units they produced in the meantime.
There's also a glitch that I don't believe has been patched whereby if you destroy a luxury good improvement (thereby reducing the amount of that luxury good available) you can end a trade agreement early at no penalty, and then rebuild the improvement and sell the good back to that same Civ for another sum of money. This could be considered cheating, and I imagine someday it (will be/has been) patched however.
Roads between cities are generally profitable, but you want to try to minimize the number of redundant road links you have, as having 2 roads between 2 cities counts the same as having one, and costs you extra.
Station an inquisitor/inquisitors in your city/cities and he will be unable to act. Alternatively you should be able to surround the city with 6 workers/other non-combats (triangle units) and he won't be able to get close enough to use his ability.
If he does successfully convert some citizens, you can use an inquisitor to remove his influence, but its preferable to block him by just stationing the inquisitor there.
I don't believe any of those will cause a diplomatic hit. If one of them does, it would be using the inquisitor to remove influence after the GP is used on a city.
Best Answer
One thing to keep in mind is that any major religion is superior to no religion at all. Yes, it gives a bonus to your opponent, but it gives a larger bonus to your city. Therefore, its advisable to use these GPs to spread religion to cities that do not have, and are unlikely to get, Shinto. Island colonies far from other cities are the best example. They would not otherwise get a religion, so it can be wise to spread buddhism or confucianism there.
Finally, look at the benefits provided by those religions. If you have a specialty city, it may be useful to add a specialty religion. For example, if you have a pure production city, and Buddhism gives the 15 percent production bonus, it may be wise to spread Buddhism there in order to get said bonus.