How does the exchange generate the addresses and assign to new users? What sort of wallet does it use?
Also if I check on the Blockchain, will my address in the exchange be there? Will it continue to show the changing balances as I transfer from one account to another in the exchange? I noticed thag the exchanges keep changing the deposit address during a period of time.
First of all: I've never seen the internals of an exchange, hence all I tell is what I'd expect to see in there. Probably some parts are simplified because I don't know whether they integrate some automation in the handling of their cold storage, for example.
For Bitcoin they probably use Bitcoin Core because I don't know any other wallet that can handle such load. They probably use it with a software built on top of it (probably something custom, but similar to what I'd expect from an e-commerce package). They have to use many more wallets to cover all the coins they operate with, for allowing deposits and withdrawals.
At the request of the exchange platform the wallet can create a new receiving address and hand it back. The exchange database will contain an information for each user (and each chain/coin) : a list of addresses (it has to be a list because if, for some reason, you are allowed to change address, the old one should still work for deposits).. And the wallet or a software on top of the wallet will monitor each new block. When a transaction comes to one of the monitored addresses, the exchange knows; when the transaction reaches a certain number of confirmations, the exchange software will credit the user's account with the corresponding amount.
So yes, your deposit address is a valid address on the blockchain and you can see on any block explorer all the transactions made with it: your deposits and also withdrawals of the platform, probably triggered by other users.
When you transfer from one account to another inside the platform, as already said, you don't transfer money from that address, you transform "numbers" from the exchange's database (the money your account was credited with). So no, if handled correctly, in-house transfers will not be visible anywhere.
Changing the deposit address is a good practice and helps for your own privacy. As I wrote, if handled correctly even the old deposit address should work (but please don't count on that, read what the specific exchange tells). And as I wrote, it doesn't matter what's the deposit address, since after the deposit has the number of confirmation it no longer matters what has happened on that deposit address (probably the funds will be withdrawn for other users), since your account is credited (the amount = number is written elsewhere in their internal database, related to your account).
