4 updated answer with better info
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Just untilReading the information from the Russian Consulate website linked above, it states that you check outdo not have to register unless you go to a different region of Russia (like if you leave Moscow for Krasnodar or somewhere far). You shouldn't have to re-register at the Moscow apartment if the Moscow hotel already registered you, but I would play it safe and re-register anyways at the apartment. They want to be able to track you down and find you, in case of emergency or if they need to deport you. Technically, your visa registration is supposed to be returned when you depart Russia, but they do not enforce that strictly.

Just until you check out of the hotel. They want to be able to track you down and find you, in case of emergency.

Reading the information from the Russian Consulate website linked above, it states that you do not have to register unless you go to a different region of Russia (like if you leave Moscow for Krasnodar or somewhere far). You shouldn't have to re-register at the Moscow apartment if the Moscow hotel already registered you, but I would play it safe and re-register anyways at the apartment. They want to be able to track you down and find you, in case of emergency or if they need to deport you. Technically, your visa registration is supposed to be returned when you depart Russia, but they do not enforce that strictly.

3 added NYE info
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If you stay anywhere (other than a hotel) shorter than 7 working days, you do not need to register. For instance, I went to Moscow for my brother in law's wedding, but was only there for 2 days. I did not register anything (but I did not stay in a hotel, I stayed with my wife's family at their home). We left for Greece for a week, then came back to Moscow for a few more days. I did not register then either, as I left Russia and went back to the USA. I never stayed 7 consecutive working days, so I avoided registration entirely. I also did not stay in a hotel. A hotel makes the registration process very easy, whereas staying with my in-laws would require a trip to the local post office and waiting in line for over an hour to register. Not fun.

Just until you check out of the hotel. They want to be able to track you down and find you, in case of emergency.

Also, keep in mind, the 7 day period is "working" days, not weekends. Also, if you plan to travel during New Years (like many of us do), keep in mind that there are no working days from Jan 1 - Jan 8th, in observance of Russian New Year and Christmas. I usually visit Russia during New Year and do not have to register because of the holiday!!

If you stay anywhere (other than a hotel) shorter than 7 days, you do not need to register. For instance, I went to Moscow for my brother in law's wedding, but was only there for 2 days. I did not register anything (but I did not stay in a hotel, I stayed with my wife's family at their home). We left for Greece for a week, then came back to Moscow for a few more days. I did not register then either, as I left Russia and went back to the USA. I never stayed 7 consecutive days, so I avoided registration entirely. I also did not stay in a hotel. A hotel makes the registration process very easy, whereas staying with my in-laws would require a trip to the local post office and waiting in line for over an hour to register. Not fun.

Just until you check out of the hotel. They want to be able to track you down and find you, in case of emergency.

If you stay anywhere (other than a hotel) shorter than 7 working days, you do not need to register. For instance, I went to Moscow for my brother in law's wedding, but was only there for 2 days. I did not register anything (but I did not stay in a hotel, I stayed with my wife's family at their home). We left for Greece for a week, then came back to Moscow for a few more days. I did not register then either, as I left Russia and went back to the USA. I never stayed 7 consecutive working days, so I avoided registration entirely. I also did not stay in a hotel. A hotel makes the registration process very easy, whereas staying with my in-laws would require a trip to the local post office and waiting in line for over an hour to register. Not fun.

Just until you check out of the hotel. They want to be able to track you down and find you, in case of emergency.

Also, keep in mind, the 7 day period is "working" days, not weekends. Also, if you plan to travel during New Years (like many of us do), keep in mind that there are no working days from Jan 1 - Jan 8th, in observance of Russian New Year and Christmas. I usually visit Russia during New Year and do not have to register because of the holiday!!

2 answered questions directly
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If you stay anywhere (other than a hotel) shorter than 7 days, you do not need to register. For instance, I went to Moscow for my brother in law's wedding, but was only there for 2 days. I did not register anything (but I did not stay in a hotel, I stayed with my wife's family at their home). We left for Greece for a week, then came back to Moscow for a few more days. I did not register then either, as I left Russia and went back to the USA. I never stayed 7 consecutive days, so I avoided registration entirely. I also did not stay in a hotel. A hotel makes the registration process very easy, whereas staying with my in-laws would require a trip to the local post office and waiting in line for over an hour to register. Not fun.

It is usually easier to have a hotel do your registration, as they are familiar with the process and usually have English speaking people to fill it out for you, plus they have all the necessary forms, stamps and paperwork support. You simply just sign the forms after they stamp them and keep them in your passport. Do not lose those forms! I have never been asked to produce them, but it could happen and is perfectly legal for the Russian authorities to request it without notice. I usually keep a low profile and keep my English speaking to a minimum, to avoid any checks like that.

Is it possible if I booked a hotel stay for 3 nights for the visa registration and proceed to stay in a cheaper apartment?

You will still need to re-register at the apartment, assuming you'll be there longer than 7 days (you said 3 weeks).

Will my visa registration last for the whole period Im staying in Moscow or is it just until i check out from the hotel?

Just until you check out of the hotel. They want to be able to track you down and find you, in case of emergency.

If you stay anywhere shorter than 7 days, you do not need to register. For instance, I went to Moscow for my brother in law's wedding, but was only there for 2 days. I did not register anything. We left for Greece for a week, then came back to Moscow for a few more days. I did not register then either, as I left Russia and went back to the USA. I never stayed 7 consecutive days, so I avoided registration entirely. I also did not stay in a hotel. A hotel makes the registration process very easy, whereas staying with my in-laws would require a trip to the local post office and waiting in line for over an hour to register. Not fun.

It is usually easier to have a hotel do your registration, as they are familiar with the process and usually have English speaking people to fill it out for you, plus they have all the necessary forms, stamps and paperwork support. You simply just sign the forms after they stamp them and keep them in your passport. Do not lose those forms! I have never been asked to produce them, but it could happen and is perfectly legal for the Russian authorities to request it without notice. I usually keep a low profile and keep my English speaking to a minimum, to avoid any checks like that.

If you stay anywhere (other than a hotel) shorter than 7 days, you do not need to register. For instance, I went to Moscow for my brother in law's wedding, but was only there for 2 days. I did not register anything (but I did not stay in a hotel, I stayed with my wife's family at their home). We left for Greece for a week, then came back to Moscow for a few more days. I did not register then either, as I left Russia and went back to the USA. I never stayed 7 consecutive days, so I avoided registration entirely. I also did not stay in a hotel. A hotel makes the registration process very easy, whereas staying with my in-laws would require a trip to the local post office and waiting in line for over an hour to register. Not fun.

It is usually easier to have a hotel do your registration, as they are familiar with the process and usually have English speaking people to fill it out for you, plus they have all the necessary forms, stamps and paperwork support. You simply just sign the forms after they stamp them and keep them in your passport. Do not lose those forms! I have never been asked to produce them, but it could happen and is perfectly legal for the Russian authorities to request it without notice. I usually keep a low profile and keep my English speaking to a minimum, to avoid any checks like that.

Is it possible if I booked a hotel stay for 3 nights for the visa registration and proceed to stay in a cheaper apartment?

You will still need to re-register at the apartment, assuming you'll be there longer than 7 days (you said 3 weeks).

Will my visa registration last for the whole period Im staying in Moscow or is it just until i check out from the hotel?

Just until you check out of the hotel. They want to be able to track you down and find you, in case of emergency.

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