How to couchsurf on a visa in Russia?

@Ach's comment is the best advice, to get a voucher from a tourist agency. I have used Apon service to do it in the past, although I have never "couch surfed".

Apon service has a website for the tourist invitation letter/voucher.

Once you arrive you will need to register your tourist visa.

What I would like to do is apply for the visa and when I get the okay I organize a couchsurfing place. How would that work?

Yes, that works just fine and is perfectly normal. A lot of people will get their visas and then change their accommodations. The letter of invitation is a formality (a silly one) but it's used to keep track of you. Once you register your visa after arriving, as long as you do not leave the region (for instance, as long as you do not leave the Moscow area), you technically do not need to re-register. They suggest you do, if you stay somewhere longer than 7 days. So if you find different places to couch surf, you should be nice and register your visa with your hosts. Keep in mind that a hotel can easily register you and has all necessary forms and knowledge to do so. A couchsurfing host will not have the forms or stamps needed, so they will need to take you down to the local post office to register you, which can be time consuming. My wife registered me once and it took almost 2 hours and a lot of confusion on the post office's behalf. As long as you register once, and stay within Moscow, you will be fine, as long as you can produce your couchsurfing details upon request. Let's say you register at a hotel first, and then after one night, start couch surfing instead. That's technically fine, but they would want you to re-register from the new place, or keep your receipts handy in case you get stopped by police. Police will check your registration if you are stopped by them.

Could I book at a cheap hotel and specify that as my accommodation and then just cancel that booking?

Yes, that's probably your best approach. Get the hotel to register you the first night you are there. Then couch surf the rest of your trip. The hotel will get you registered within 10 minutes of you asking them. A couch surfing host could take a much longer amount of time. Just keep in mind that they suggest you re-register any place you go, just as a formality, unless you'd rather keep all your receipts and document handy while you walk around the city.