This site is only useful if you look at the charts yourself. Their suggested fee at the bottom is usually way too much unless you absolutely need to get in to the next block. At the moment their suggest fee is 22 sats/byte, which would put you around 0.05 MB from the tip of the mempool.
Agreed, and edited my post.
To answer your other question, yes you can include a very small fee and it might take a long to for your transaction to confirm if it ever does. After several days, most nodes will forget your transaction if it doesnt confirm but it will remain valid as long as all the inputs remain unspent. This means someone may broadcast your transaction long into the future if it is still valid. This means you should keep your private keys, including backups if you ever even try unsuccessfully to send coin between wallets.
Wow, so there is no way to cancel an unconfirmed transaction even after say 2 weeks of it being unconfirmed? What happens then? How to get it on the blockchain later?
If you spend one or more of the inputs from the original transaction, the entire original transaction will become invalid. If the original transaction does not confirm after a couple of days, most nodes will 'forget' about your original transaction and will accept a new transaction with the same inputs, but if you do not spend one or more of the inputs, the transaction will forever remain valid.
This really only matters if you send coin to a new wallet, the transaction doesn't confirm and you lose the private keys to your new wallet.