Does sending coins around within your wallet have the same privacy effect as sending them to your other wallets?
Also, Why is there no transaction fee required for this?
i think i read somewhere, that it is looks obvious on the blockchain (less private)?
Sending BTC within your wallet or to other wallets really doesn't give you any privacy effect. The coins can be tracked using taint analysis. If you need privacy you need to mix the coins using a mixing service. CoinJoin is "pretty" good and is trustless.
Sending from one wallet to the other or within the same wallet requires the same tx fee.
Also, Regarding privacy, is it even better to create a new electrum wallet (instead of just new address) for each incoming transaction?
Not really. If you use coincontrol properly it makes no difference whatsoever.