Even if we enter the transaction fee too high, it will be confirmed at the earliest 10 minutes.
The 10 minutes is an average. Just look at https://blockchain.info/blocks and see the times between blocks. In order to have a payment confirmed, a new block needs to be mined, so there are the real times.
Sometimes it's less than 5 minutes, sometimes it's more than 20 minutes.
No, it doesn't. I use this site to see: https://btc.com/stats/unconfirmed-tx
The recommended fee now is 50 sat/byte. An average value would be some 25 sat/byte, making your transaction get confirmed in about 12h. It's quite a leap from ~10 minutes. And it's an estimation.
If the network gets very busy suddenly, many transactions will come in after yours but with higher and higher fee. Yours will remain unconfirmed for longer and longer. And lower fee you use, higher chance this to happen to you, especially if you don't check first how full the mempool is (how busy the network is).
If the value is too high, confirmation should take as fast as the next block or current blocked being mined takes to be mined.
https://en.bitcoin.it/wiki/Transaction_fees writes: