Black hole radiation in which particle and antiparticle pairs are separated which ultimately leads to the 'evaporation', but why does the negative particle always enters the black hole? Why do the particles appear at the first place? There should be an energy source for the same ?
Before anything else, it should be pointed out that the virtual particle pair is not a detailed description of the process, but rather an analogy that Hawking came up with in his original paper describing the process to gain some intuition about it. Quoting Hawking himself,
It should be emphasized that these pictures of the mechanism responsible for thermal emission and area decrease are heuristic only and should not be taken too literally.
In fact, any attempt to describe the process in full detail using particles close to the black hole will be flawed, since it is not possible to define what is a particle when close to the black hole. It should be recalled that this is a prediction of quantum field theory in curved spacetime and that quantum field theory is a theory of fields, not particles. Particles are useful in some contexts, but are not fundamental.
That being said, let us try to address the issues you raised. I'll change their order a little bit so the answer is easier to follow. In order to address the issues raised on the comments, I'll then write a little bit on how the effect is actually derived.
Addressing the issues with virtual particles
Where do the particles come from?
Firstly, where do virtual particles come from? Within quantum field theory, virtual particles are a way of interpreting the quantum fluctuations of the vacuum. In short, the vacuum is not "empty", but rather it is subject to quantum fluctuations that can be interpreted as the creation and subsequent annihilation of particles.
Shouldn't there be an energy source for the particles?
For the remaining points, I resort to the original explanation that Hawking gave in the analogy he presented in the original article. As you mentioned, the result is interpreted as the creation of a pair composed out of a positive energy and a negative energy particle, so that they both sum zero energy. Hence, there is no issue with energy conservation.
Why doesn't the positive energy particle fall into the black hole?
They are originally created as a pair of virtual particles, which must be annihilated shortly. However, if the negative energy particle tunnels to inside the black hole, it will behave as a real particle (in short, because time and space change places inside a black hole), allowing the two particles to go off in different directions and one has radiation. This can't happen if the negative energy particle doesn't fall into the black hole, and hence it is necessary for the negative energy particle to fall in.
In short, the particles can't exist as real particles if the negative energy particle doesn't fall into the black hole. Unless it falls inside, they must annihilate. If the positive energy particle falls together, then they can still annihilate and nothing changes. They can't exist as particles if only the positive energy particle falls in (more on this later, please keep reading).
Why is the energy of the black hole depleted?
Because it is absorbing the overall negative energy of the falling particles.
These explanations seem a bit wavy and weird...
That's because they are. Virtual particles are not the source of Hawking radiation. They are merely a cartoonish analogue that Hawking came up with to allow us to obtain a better interpretation of the process, but it should always be taken with a grain of salt. Any depiction of Hawking radiation in terms of virtual particles will always have questions that can't be answered well enough (I believe my answer to "Why doesn't the positive energy particle fall in?" might have left you unsatisfied) because virtual particles are not the source of Hawking radiation.
So what is the origin of Hawking radiation?
The fact that particles are not a fundamental concept of nature. They are a useful tool in many contexts, but they are just as fundamental as our predated notion of absolute time in Classical Mechanics. Time is relative, and so are particles.
The notion of particle depends on what an observer calls positive or negative energy. However, it is not straightforward to define this in General Relativity. Energy is defined as the conserved quantity associated with time-translation invariance by means of [Noether's theorem]. This is a rather technical way of saying "Energy is a number conserved due to the fact that the system does not change with time". However, in Relativity, the system very often does change with time, and many times we can't even pinpoint what exactly we mean by time. This leads to the consequence that we don't really know what to call energy, much less what we should call particles.
When I said particles are just as relative as time, I meant it quite literally (although there are still some subtle differences).
In some spacetimes, known as stationary spacetimes, there is a preferred notion of time. For example, suppose you have an spherical star that is perfectly stable. We often can consider it to be stationary and understand that the gravitational field won't change with time. However, suppose that for some reason it undergoes gravitational collapse and forms a black hole, that eventually settles down into an "equilibrium state", so that it doesn't change anymore. It is now stationary.
The trick is: this story is stationary in the past and it is stationary in the future, but not in the middle. And due to this, the notions of time in the future and in the past are different.
And if the notions of time are different, then so are the notions of particle.
What happens in the Hawking effect is that what an observer calls a particle changes after the black hole forms, and this leads to different predictions. This leads to the detectors reacting differently. And this is just as wild as it looks like. It puzzled quite a bunch of people who decided to dive deeper in the problem and they found out that these sorts of things happen in even less extreme scenarios: the Unruh effect is the prediction that when an inertial observer is in vacuum (i.e., not seeing any particles), an accelerating observer is seeing a thermal distribution of particles with temperature proportional to the acceleration. The mere fact that the observer is accelerating changes the notion of time, which changes the notion of particle.
In short, particles are not absolute. They are observer-dependent.
Then how come Physics?
We don't need particles to do Physics, we only needs the fields. The particles arise in specific situations in which they admit useful interpretations, but they are by no means fundamental. Apparent paradoxes can arise (If the accelerating observer detects a particle, then how can the inertial observer explains it if they see no particles?) but they can always be solved (the inertial observer attributes it to the vacuum fluctuations).
For a different text discussing the same issues, you might be interested in reading one of Sabine Hossenfelder's blog posts.