It is an unrealized gain if you bought it and it appreciates. If you receive it in exchange for services, it is income. This is the same for non-crypto:

1. You buy an artwork by a famous artist for $10k and it appreciates to $20k. This is an unrealized gain and you are not taxed until you sell it. Note that trading one crypto for another is effectively a sale of the first crypto and a purchase of the second, such that you pay gain depending on the value of the second crypto at the time of the trade.

2. Someone gives you a $10k artwork as partial payment for some work you do, eg web design. This is income and taxed as if they paid you $10k. Mining crypto is also income I believe.

3. Someone gifts you an artwork worth $10k. You don’t pay tax.

Etc.

I would think the best way to characterize this transaction is that it is income, although it’s certainly unusual enough that I won’t say I’m certain.

Not tax advice / speak to your own accountant.

In point 1, why does trading of crypto incur a taxable event? If I buy a Picasso for $20 million and then later trade it to a friend for a Van Gogh which is worth $40 million, is that taxable?

Yes it most certainly is.

In this scenario, what is the taxable base ?

You would need a valuation for the painting you disposed of. The fact that you never actually received USD is irrelevant.

I had this problem in 2018 - I bought BTC in 2017 before the boom, exchanged a bunch for ETH at the peak of 2018, and then rode the ETH all the way back down. I had to pay a bunch of taxes on the BTC gains I “realised” even though I didn’t get any actual cash out. I had to pick a price settle for BTC/USD on the day I bought ETH.

Same in French law if you move a company abroad: You need a valuation on the day of departure, and are taxed on it, even if you don’t realize the gains.

If you receive it as income then it's taxable at that point at $ value. If you later sell or trade it you realize capital gains (or losses) and that is a further taxable event.