1. That you can trade anonymously with cryptocurrency
The fact that cryptocurrencies can be bought and traded, with anonymity, is a truth with modification. A transaction in a Blockchain is always traceable.
At bitcoin.org writes that: anonymity has not been a target of cryptocurrency. You have some anonymity until the day you make a transaction (trading something for their cryptocurrency). But once you have made a transaction there is traceability. For example, when you buy a trip from Virgin (which, for example, receives Bitcoin) or a computer from Microsoft (which also receives Bitcoin) or buys something with cryptocurrency (usually Bitcoin so far), you can track who has traded, which you can link back to where you live, what transactions you previously made, etc.

2. The use of cryptocurrency for illegal trading
There are no statistics to indicate that there is a higher frequency of illegal trading in cryptocurrencies compared to, for example, Swedish kronor or euros.
One of the biggest myths about cryptocurrency is that it is primarily used for illegal trading.
The first of these is that it is "widespread" that cryptocurrency is used in shady deals such as in drug trafficking, terrorism, and more. The list is long that cryptocurrencies are used to buy and sell banned goods, that they are used for illegal tax fraud, transactions, money laundering, because of its alleged anonymity. Anonymity with modification.
In all types of trading with all types of currency, there is always trade that is not legitimate. But as in the cryptocurrency as well as the "fiat" currencies, it is usually possible to track.

3. That you can use cryptocurrencies to avoid tax
Can you use cryptocurrency to avoid taxes? No. For example, the British Tax Agency has concluded that Bitcoin (one of many cryptocurrencies) cannot be equated with foreign currency. If you trade in it, that capital should be taxed. Which means that those who made quite a lot of money in the cryptocurrency may get some surprises. Specifically, those who made a lot of money in cryptocurrency but did not declare these.