Please tell me is it now possible to send and receive BAT inside this wallet or not.
Yes, you can send and receive BAT if you are using this Crypto Wallet feature of Brave, you just need to know that their 'Brave Reward' is different on this Crypto Wallet.
(...)
by the way, weren't they asking for KYC when you used cryptocurrencies in their network?
No more KYC for this, since those KYC things about Brave browser is on their 'Brave Rewards'. For those who are able to use that feature, especially when you want to withdraw your BAT in their platform. This another feature of Brave browser which is Crypto Wallet is different for their Brave Rewards.
(,....)
you can't say it is without any extensions when the extension is already pre-installed in this "special" browser you download.
It comes with the browser, you don't need to google "wallet extension", find an extension and install it. That's what's special about this. Maybe you can think that way, but the same can be said for many other things (e.g Samsung's pre-installed wallet).
We can say that it is convenient, especially on browsers, since I think there are a lot of people has a lot of extension on their browsers, especially those using metamask which is if you use some browsers, such as Google Chrome, you need to manually install some extensions (which may be also dangerous if you installed a wrong one).