I know that there are huge transactions in each bitcoin wallets. Different security levels, functions, difficulties.. But there is something bugging me, if I have an for example a BLOCKCHAIN account then I wanted to transfer my funds from Blockchain to my Local Bitcoin Wallet, is it possible?
Yes.
And if its possible, do they have processing fees or whatsoever?
As far as the Bitcoin protocol is concerned, all fees are voluntary.
Bitcoin is a voluntary system for everyone involved.
You are not REQUIRED by the protocol to include a fee.
The peers on the network are not REQUIRED by the protocol to relay your transaction to any other peers on the network.
The miners and mining pools are not REQUIRED by the protocol to confirm your transaction if they don't want to.
Miners (and mining pools) have to spend money to purchase and operate their equipment. Therefore, they prefer to receive some payment to cover those costs and earn a profit. If you voluntarily include a fee with your transaction, then a miner will have an incentive to include your transaction instead of somebody else's transaction when they are building their block of transactions to confirm. If you include a high enough fee, then you will provide enough of an incentive and some miner (or pool) will confirm your transaction for you.
Some wallets will allow you to send transactions without any fees (or with extremely low fees), and leave it up to you to decide if you want to take the risk that the transaction takes a VERY long time to confirm. Other wallets creators protect their users from accidentally taking such a risk and will compute a fee and force the user to use a fee of at least the computed amount. That all comes down to which wallet you choose to use.
The most popular wallets don't typically collect any fees for the wallet creator, but if a wallet creator wanted to they could create a wallet that pays the wallet creator a small fee. There may already be a few wallets less popular wallets out there like that.
And, or, I should try to consider each bitcoin wallet their fees? Their service?
Yes, before choosing a wallet, you should consider all the features of the wallet and determine which one best suits your needs.
I hope their fees would not eat all of my hard earned btcs.. Thank you!
That's up to you, and how you choose to use bitcoin and the wallet.
Since bitcoin transactions must spend each received transaction output separately, the more tiny value transactions that you receive the larger any transaction that spends those bitcoins will be in terms of bytes. Larger transactions (more bytes) means you'll need to include a higher value fee to provide enough incentive for the miner (or pool).