"If Earth was on fire after the Big Bang, how did the first cell develop in such harsh conditions?"
First, this has nothing to do with evolution. The theory of evolution is a model which describes how the diversity of species arises. It includes two components, variation within populations, and selective pressures. Importantly, it is a model which describes biological systems, not physical ones. How the Earth formed is not a part of the Theory of Evolution.
The formation of the Earth is a process understood in the context of planetary formation. Planets are believed to form from the remnants of their host stars, which are formed via gravitational collapse of large clouds of gas and dust. The process has been extensively modeled, and observed directly.
While it is true that the Earth was molten, as it dissipates both heat from radioactive isotopes and heat from accretion, the Earth cooled down over long periods of time. Life is believed to have formed after the period of time in which the surface conditions would have prevented it from doing so.
The Big Bang is an unrelated process, which is the observed mechanism by which large scale structure forms in the universe. The Big Bang is directly observable through several independent lines of evidence.
"Did humans crawl out of a swamp? How did we evolve from a water-based organism?"
The transition of life from the oceans to land is a very interesting topic. Here is a good article about the origin of tetrapods.
"If everything started off from a Big Bang, how is Earth in mathematically perfect distance away from the Sun that allows survivable conditions for all living things?"
Again, the Big Bang has very little to do with the details of the formation of the Earth or the development of life. The Earth formed through accretion from materials remaining after the formation of the sun. The habitable zone of our star is enormous, stretching from Venus to Mars. The concept of a habitable zone, itself, is questionable. Life may exist in radically different environments, including the moons of the gas giants and on smaller bodies throughout the solar system. Leaving that aside, the Earth is where it is, specifically, because of statistics. There are trillions of galaxies containing billions of stars and trillions of planets. The process which forms planets is stochastic. Some will be within the habitable zone of their star.
"Is it all just a coincidence that there is no other place like Earth in rest of the Milky Way galaxy as far as humans know?"
No, it isn't a coincidence. We have only had the technology to look for Earth like worlds for several decades. If you'd like to see Earth like planets discovered, consider donating to the planetary society., an organization which lobbies for funding to do these kinds of surveys. Estimates of the number of Earth like planets in the Milky way are in the billions. Perhaps as many as 1 in 5 sun like stars hosts a planet like the Earth.
"Why is a human body such a perfect creation with no flaws--from toes to brains?"
Humans are hardly flawless.
"Why do humans have conscience from what is wrong and what is right?"
Because we are social mammals.)
"Everything is so perfect, it can not all be just a coincidence. There must be a higher power (God or aliens) at work."
I'd advise your friend to spend more than just a moment considering how "not perfect" the world is. More than a billion people live in poverty. 150+ children were murdered in cold blood in Pakistan today. Even if you ignore how evil people are to each other, 99.9999999% of the universe is inhospitable to life as we know it. The only places that we know of which are hospitable to life are the surfaces and near subsurfaces of small rocky worlds - a minority of the things and places in the cosmos.
I'm having hard time convincing him, and I hope you guys change his mind. Can you please also provide proof and your credentials so he will listen?
What people can provide is evidence and rational argument. Credentials are a proxy for those things. If he/she accepts arguments only on the basis of authority, what they need is a lesson in skepticism and a bit of maturity.