Security implications of regular saving into cold storage
Q1: Depens on how you use it. 
Transactions to an address does not decrease it's security.
Using a cold-storage wallet and taking it online exposes your wallet and it's security.

Q2: It's correct. Most exchanges require KYC(know your customer), they store information about you and your withdrawal addresses. It's possible to create a new address for each withdrawal. It's hard to prove that you are in possession of the withdrawal address.


  (the reason why I want to do 2 is that I do not feel I really own bitcoins until they are in a wallet under my control)


Can't stress this part enough, you only own bitcoin if only you own the private keys.