Booked two separate accommodations after getting a visa, is it a problem?

It is not a problem to have multiple accommodation arrangements for a single country as long as there is coverage for the entire period and the arrangements dovetail with your itinerary.

If you already have the visa and have cancelled your original reservations you can prepare a revised itinerary and attach the booking confirmations. If (when you land) the immigration officer wants to see it then you can show it and provide a verbal explanation about why there's a variation.

For your other question about whether an apartment rental through airbnb is acceptable accommodation: yes, it's fine as long as the confirmation is in order.

Overall, you should bring along the documents you used to apply for your visa along with an updated itinerary and new confirmations.

List the first address in the immigration form. Generally the officials don't care much about the address. Airbnb is fine as long as you have the receipt to show the booking.

I like Gayot Fow's answer, and that's the safe strategy. That said, I have done modifications like this and more, and was never asked for the reservations etc while traveling.

There are two considerations to this, which are formally the issue. The enforcement of them (if any) varies, depending on country you are from and country you are traveling to.

Hotel cancellation as proof of invalid application

Schengen visa can be revoked if you cancel the hotel booking, submitted with application. This had just made news recently here (Ukraine) by becoming a little more common for us lately.

The reason for it is primarily people using “easy” countries to get visa to get into “hard” countries. Canceling your hotel can be interpreted to indicate you never intended to travel to the country you applied for (regardless of it being Schengen visa).

The recommendation (as covered by news) is to email embassy about changes like that in advance.

Hotel registering your stay with authorities

I hadn't looked into it extensively, but to my knowledge plenty countries require visitors to register with authorities for anything but very short stay.

This is typically handled by hotels, which is (in part?) why in some countries they require your passport presented for check in.

If you have alternate accommodations you might be legally obliged to report your stay yourself.

I don't think this is particularly strictly enforced, but it is also a consideration.