Is there any difference or any problem or extra fees for having permanently (1 month, 6 months, etc..) my BTCs instead of having them in a local wallet like Bitcon-Qt or Multibit?
This really depends on the exchange website's policies. But in general money is a local wallet behaves just as what a wallet in real life would do: store money without any fees or interest. As long as you (and only you) own the wallet file (normally
wallet.dat) you control whats being done with the money.
On the other hand, money stored on a website/exchange behaves very much like storing money in a bank. Fees or interest is based primarily off the website's policies. The main thing thats different is that they reserve the right to do things with your money. When you are sending money to a address they merely carry out your request for you.
Regarding long term storage a local wallet should retain all money you have as long as you own the wallet file and that no major changes to the bitcoin system itself occurs (extremely unlikely). For online exchanges, it depends on whether they choose to do anything with your money whilst you are not using it (e.g. remove it after x amount of inactivity). Needless to say if the website/exchange shuts down for whatever reason you probably can't get the money out again.
Given that a exchange/website employs a no-delete, no-cost policy it really just boils down to trust. Do you trust a third party with your money for whatever the benefits it may bring you or not?
Only the respective trade-offs of trusting another party with your money vs trusting yourself not to lose your data.
When you keep your money on an exchange website or online wallet service, you are not only trusting their good intentions, but also that they stay in business and are running a secure service. A significant number of exchanges and OWS have been compromised or put out of business.
When you keep your money on your own devices, you are trusting yourself not to be hacked, fooled into sending money to a wrong address, and to keep your data safe. The latter requires you to engage in a decent backup policy and to have a basic understanding what you are doing. There have been numerous cases of personal computers being compromised (google for example "AllInVain"), people losing their data to hardware failure without having a backup, or recently, someone just throwing his hard drive worth $7Mio in the garbage.
First let us know the difference between the private bitcoin wallet and Third Party wallets. For bitcoin wallet the private key of the wallet is the key to your bitcoin wallet, so anyone who own/know can easily spend any money in that private key any time.
In private bitcoin wallets the private key is known to you only and you don't share with anyone else. It is more secure than a bank vault if executed properly. Here you are responsible for saving and securing the funds in your wallet and creating the public address for transactions etc. There are many offline wallet options like, desktop wallet client, a mobile offline wallet, any Paper bitcoin wallet or offline wallet hardware. There are no charges associated directly with
In Third party bitcoin wallets the private key of your account is held by the website, and most frequently they just show your bitcoin balance as reflected in their database and they use some cold-storage wallet to store the actual bitcoins of all the users at their site. Most of the time you can create only one or two public addresses, and all the payment to that address is associated your account. This online wallets generally charges on the basis of transaction value, the charges vary from 0.01% to 0.1% BTC per transaction. Most probably that don't charge you based on the time of duration you keep them in the wallet.
note: some Bitcoin wallet sites can delete your account if you don't access your account for some pre-defined time like 3-6 months. so read terms carefully.
I think another difference besides what have been said is, when you put money in an e-wallet, the third party can accumulate a decent amount of money from all the users, and use it for short term investments. That's why you can use their service freely, or even get some interests. This is what I see in e-wallets like "WeChat pay" or "Alipay". However with bitcoin wallet, you are keeping your money in your own pocket all the time, and to use it you need to pay a so called "small" amount of fee.