If I’ve moved my ADA to cold storage (e.g. ledger nano) having previously defined / chosen a specified stake pool, where do any staking rewards go?
Not sure 100% if it goes to your address or you create an address for rewards …
You can’t move your ADA, whether to cold storage or anywhere else, while keeping it staked. It has to be staked from the wallet in which it is going to stay. There will be special arrangements for staking from cold wallets.
Are you sure ? I have read / heard somewhere earlier that coins can be transfered freely despite stake. It is only important what stake you have at the time of settled time frame. Eg. at the end of each epoch you will get a divident according to your stake at that moment.
Yes the coin could be transferred freely during staking …
My understanding is that staking does not prevent movement, but movement stops staking (at the end of the epoch). Coins have to be restaked after being moved. If you move them into a cold wallet they then have to be staked using a special procedure, regardless of whether they were staked previously or not (but staking does not actually start until the beginning of the next epoch).
Not necessarily. You stake specific address (wallet) So it stops staking for what you moved out but what is left still on that address gets reward. Of course at specific time frame. This is my understanding but needs confirmation. Moved coins need restaking.
I said “movement stops staking”. I don’t know why you would think I meant for all coins.
I have not clearly read to the end of your thought , my fault, but I think we’ve come to the likely scenario at the end. What is at stake is profited what moved needs restaking 
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Just reversing the question here,
If I have staked an address then move additional ada to this staked address will it automatically stake ?
I think if it is before the limit date then yes
you never know if your stake wins the epoch
I believe a snapshot is taken at the beginning of the epoch.
If a cold wallet is created, a staking key is also derived from the private key and saved in the wallet software. This staking key allows to delegate ADA but it has no control over the funds. (like the private key)
Staking rewards go into a new “reward address” tied to this staking key. It’s a very special address, not UTxO style like the rest, but account style address to prevent something that is called “dust” and bloats the wallets. Of course it will just show up in your wallet, so basically you don’t have to care about this detail at all.
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Thanks if you are looking for another topic for a video, this might be one to consider…the staking reward journey of ada.
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Yes it will.