1)It is advised that to have the perfect security, you should remember/write down the private keys of every bitcoin address where you receive funds, right? At the same time, it is advised not to use an address to receive funds more than once. So do I really need to write down every private key of every bitcoin address I receive funds to? seriously? I know this might be overreacting, but I really want to have my wallet 100000% secured.
Electrum is a deterministic wallet which means that all address specific private keys are derived from a single seed. This is shown in the user interface in the human friendly form of 13 random words. The words and their order is important. Write down the seed and it is the only backup you need. *All* the address specific private keys can be derived from it. The addresses themselves are derived from the private keys so all your addresses can be recovered with just the 13 word seed.
If you like you can also make a backup of the wallet file using file menu > save copy. The wallet file stores address and transaction labels that cannot be recovered from the seed alone.
2)Upon creating a wallet in electrum, I am provided with 10x different addresses, but I can't seem to be able to create more. Now if I do not want to use an address more than once, 10 addresses is definitely NOT enough for me, I fail to see any button or something to generate a new address. Also, if there is actually a way to generate more addresses, will these new ones also be backed up by the seed?
Electrum will automatically create new addresses from your seed when you start to use the old ones. It tries to make sure you always have at least 10 UNUSED addresses. As mentioned above all addresses and address specific private keys can be recovered from the seed or a one time wallet file backup. Although if you add a lot of labels you will want to make more frequent backups of the wallet file just for the labels.
Oh and desktop wallets are definitely better than web wallets.