Is flipping a coin really 50/50 or is it 49/51?

As a trope, flipping a coin means a 50–50 chance of heads or tails.
The exact proportion of heads and tails depends on the coin and on the method of flipping. For the usual flipping by hand, a coin has a slightly greater chance (about 51%) of landing on the same side as it started on. But that is for a perfect coin. The degree of bias depends on the number of flips it goes through in the air (more flips are less biased) and whether it is caught in the hand or let fall to the floor (falling to the floor is less biased)
A coin that is spun has a huge bias favoring the heavier side landing down.
Also, some people can flip a coin and control how it lands; this doesn’t involve any esoteric powers, just a lot of practice flipping coins - it’s a magic trick, not telekinesis.

Yes, but also no.
The thing is, even if you had a perfectly fair coin, it would be impossible to prove it. It’s entirely possible I could flip a fair coin 100 times and get the same result. Or maybe after some number of flips, it’s 75% biased towards tails. You just can’t know.
However, you can flip the coin many many many times, and look at the average result. This will give you a relatively good prediction for how fair the coin is. If you only flip it a few times, a 2% difference between 50/50 and 49/51 would be extremely likely.

So in the end, the question is unanswerable. Since we’re talking about statistics, you can never prove the underlying probability of the coin. You can only run tests, and use that data to predict a result. The more data you collect on coin flips, the more certain you can be. But there will always be some small margin of error. If I flip the coin 10 times and get 6 heads, you can’t be so sure that the coin is 60/40. If you flip the coin 10,000 times and get 5,003 heads, then you can be much more sure of your result.
People like to talk about “lying using Statistics” and a coin flip is a great example of that. Say I have a coin, and I want to manipulate the data to get a result. Say, for example, I want to prove the coin is 60/40. I can just flip the coin over and over. Once 60% of the flips are heads/tails, I stop. Then, I can claim the data shows the coin is unfair. This is why you should be very careful when looking at the sample size for some tests. If you flip a coin and record the result 1000 times, that’s normal. If you flip the coin and record the result exactly 815 times, that’s suspicious. That implies you did the test exactly until you got the result you wanted, then stopped the test.