Correct.  Technically the seed is just a VERY VERY big number.  Then a standardized system is used to represent that number with a set of words that are easy to write down.

When you enter the seed into a bitcoin wallet program that supports that seed, it internally converts it back into a number and then uses a standardized set of mathematical formula to generate keys from that number.  Without that starting number (represented by the seed) it is impossible to generate the same sequence of keys, so as long as you keep the seed secure, nobody else can access your bitcoins.


Correct.  As long as you have the seed, the mathematical formula used to generate keys can be repeated at any time starting at the seed value to regenerate the keys if you lose them.  You can iterate the key formula to generate as many keys as you'd like.


At the technical level, a private key is just a VERY VERY big number.  You can store it anywhere that you can store a number.  Bitcoin wallets and services are designed to make it easier to manage those keys (frequently so easy that the user doesn't even need to understand the technical details).


Some wallets and services support the Seed and HD concepts.  In that case, you simply enter your seed into the software, and the software handles the generation and management of your keys for you.  If you don't have a seed yet, then they will generate one for you and tell you what it is.

In other wallets and services, they have their own methods of managing keys.  In those cases, you might be required to trust them to manage the keys. They may not give you access to the keys at all.  Or they may provide you a method of creating a backup (generally a file that contains your keys) that you would need to securely store somewhere.


I don't know a lot about BitGo, but I don't think they support the seeds that Electrum generates.


It depends on the device. Some devices may only save the seed, and will just re-generate the keys whenever they are needed. Other devices may save the seeds as you use them so that it can access them faster.


That would be possible.  It is up to you to decide how you want to secure your access.  The two important things are:

1. You can always access the necessary keys when you need them.
This may mean accessing the keys directly, or it may mean generating them from a seed. The point is that if you are unable to access the keys at all, then you can not send your bitcoins anywhere.

2. Nobody else can access your keys.
The more copies of the keys or seed that you make, the more you increase the chance that someone else will gain access. Giving your seed or keys to a website, or untrustworthy software can result in those keys being used without your permission. Paper is nice because it can't be accessed remotely by hackers, but you'll need to import the information from the paper when you want to use the bitcoins.  If you import it to software on a computer that is infected with malware, then that malware could access the keys and access your bitcoin without your permission.  This is why hardware wallets are popular.  They keep the keys (or seed) off the computer both during storage AND while using them.


There are MANY wallet implementations.  Some of them don't use the HD concept, so each key MUST be individually saved.  Also, not all wallets use the same seed system.  The seed concept was introduced to make it much easier and safer for users to create and maintain their backups.  So, if you have access to a seed, then there is generally no need to save the keys separately.