So if you don't reuse your addresses then you will have two layers of encryption, but if you do you will only have one? But this only applies if you reuse the address after you have spent money from it.
Pretty much correct - to spend an "unspent output" (i.e. a part of your total BTC "balance") you must include the public key for the address and a script signature (so it can be verified). At that point any other "unspent outputs" to that
same address are *more vulnerable* as your public key is now known (before only a RIPEMD hash was known).
If i am understanding right, this is also related to why when you spend coins, the entire balance in that account is spent, but whatever change you need is sent back to you in a new address.
A transaction is actually a "script" that can have multiple inputs and outputs with the "inputs" being either "coinbase" (from mining) or "unspent outputs" (from transactions that were sending funds to yourself).
Each "input" must be completely spent (fees are actually the total amount of the inputs minus the total outputs so you effectively don't spend a small amount of your inputs in order to "pay a fee").
Typically your input(s) are not going to exactly match the amount you want to send and so a "change" address is added as another output to solve this.
If I spend money twice or three times, will the client or the network automatically change the account that is being spent from every time?
The network has nothing to do with it - your client will "somewhat" randomly choose which "unspent outputs" to use (with some checks to avoid you needing to pay too much in fees).