What's the deal with "legacy" and "segwit" wallets?

Question

I'm new to bitcoin, but I've already read about "legacy" and "segwit" wallets. At the beginning, I created a segwit wallet, given that it was described as a new standard. But then I found out that a market that I use can't send me bitcoin to this wallet because they use a legacy wallet. Later I read that it's impossible to transfer bitcoin between those two kinds of wallets. Is it true? If so, what's the idea? Because to me it doesn't make any sense. Especially that a new standard is probably created in order to obsolete the old standard and develop whole network using the new standard. Does it mean that my bitcoin in a legacy wallet in a certain point of time will be useless?

I tried this kind of transfer (legacy -> segwit) using Electrum and it didn't work. Does software matter?

When it comes to paper wallet created on bitaddress.org, which type is it? What kinds of problems with compatibility can it create?

Here it is said: "there is no general issue preventing a wallet from generating addresses of various output formats". But Electrum asks you what kind of wallet you want to use in the beginning, when you create the wallet. So... there must be some issue.

Answer 1

Later I read that it's impossible to transfer bitcoin between those two kinds of wallets. Is it true?

This is not true.

However a wallet needs to know about Segwit addresses in order to be able to send to them. Since it is a new address type (well new when it was introduced 3 years ago), software that hasn't implemented it yet won't know what to do with it.

I tried this kind of transfer (legacy -> segwit) using Electrum and it didn't work. Does software matter?

In what way did it not work? Electrum supports and uses segwit, so it should, unless you are using an older version of Electrum. Both the software and version of that software do matter.