It is usual / possible that a consultancy contract contains hourly rate in Bitcoin unit and not in EUR or USD? And so payer will pay via Bitcoin and not via regular bank transfer?
It is important to note that the legality of the use of bitcoins (and the recognition of them) varies wildly from country to country due to the lack of regulation of the currency and the potential for money laundering and tax evasion.
However, since there are people who have no problem working for bitcoins, it is definitely a possible mechanism by which people can be paid. So yes, if you and the other party agree for payment in bitcoins (and both people aren't breaking the law by doing so in their country), I don't think there would be an issue.
In Denmark (where I live) a chain of 200+ restaurants and fast food joints have accepted BitCoins as a valid payment, and this has made a solid shift in how danes see bitcoins. (Especially students)
So here, it's just as valid as say a PayPal transfer in EUR or USD.
But we felt the solid social differences when talking to our neighbours in Germany, where the few that has heard about BitCoins are still very sceptical.
So it is absolutely possible, I don't know of any country that has rules about it. I guess the main thing here is : Just ask.
You can make an invoice in any currency you want, even digital currencies. We had a community manager that got paid in EVE Isk, and that was completely valid with legal since it was his payment request, and not us forcing it upon him.
Unfortunately, you can't eat a bitcoin, apparently unless you're in Denmark.
You'd have to either spend all your bitcoins with merchants that accept them. Otherwise you'll be responsible for converting bitcoins back to some local currency that you can actually use, and bitcoin purchasers will charge a fee (as do all money exchanges) for you to do so. It might not be worth the hassle.
On the other hand, maybe you'd just like bragging rights to say you have some bitcoins. Whatever the case, know the true costs first.
Of course it is possible. You can simply use services like blockonomics to create invoice using your own bitcoin address. You can HODL the bitcoin received or convert some to fiat using a local exchange. An advantage of bitcoin is you can receive small amounts like 5-10USD which would otherwise require lots of fee.