Here is the situation. I am non-european but staying in Germany with a Residence permit. I can visit all Schengen listed countries with no issue. Assuming that I would want to travel to a non-Schengen country e.g. UK or USA, can I use my residence permit to travel there or am I required to apply as a citizen of my country? Assuming Germans are not required to pay Visa fees to that country, will I have to pay since I only have a residence permit? Third case, Europeans can fly directly to the airport of some countries and apply for a visa from there. In some other countries, you are required to book a Visa appointment at the embassy, which may be granted or not. Does that apply for one with a residence permit?
A residence permit does not make you a citizen, and other countries visa requirements are almost universally based on the citizenship of the traveler. So you need to apply for the same visas that everyone else with your nationality do -- except for very few and unsystematic exceptions.
The Schengen rule that a residence permit from one member implicitly works like a short-stay visa for the others is one of these specific exceptions. Another exception is that having a residence permit from an EU country can under some conditions qualify you for visa-less transit through the UK even if your nationality is one that would otherwise need it.
But unless you know that a specific exception of that kind applies to you, the safe bet in general will be that there isn't one.
You asked a general question, and the very general answer is that you have to apply as a citizen of your nation, but you will mention the residency in Germany.
The German residence permit may help to get the visa, but you still need one.
There is a short list of countries that will allow entry to people with a Schengen visa without being part of the Schengen system, e.g. Macedonia.