So was the business of paying miners fractions of existing bitcoins to verify transactions recently introduced to compensate for this difficulty, or has this option always been there?
The option (transaction fee) has always been there. Depending on the transaction size (inputs/outputs) and network congestion it varies.
https://bitcoinfees.earn.com/#fees2. In a guide to the electrum wallet I read that each time you make a transaction from your electrum waller you can choose to offer up a fraction of the amount to miners so they can verify it. But you do not get this option when you are trying to send bitcoins from a bitcoin exchange account, so where does this leave you then? Do you just have to accept that sending bitcoins from a bitcoin exchange account is going to take longer?
Not sure if the exchange you're using (haven't used HitBTC and Bitstamp) is similar to the local exchange that I'm using but whenever you're sending
BTCitcoins through an exchange, you have less control over your money and as result some options might be missing. If the exchange itself decides to choose a low transaction fee (on congested periods like the past couple of days), then the possibility of the transaction to receive a confirmation in a delayed manner is very high.
This page lists several transactions, including mine. It looks like the total value of the transactions is about 51 BTC and all of them are still unconfirmed. So what is this page telling me?
Have all these transactions been put into the same block, and is it the block itself that is waiting to be verified?
That page tells about the number of input and outputs of the transaction (aside from the self explanatory information). When you use an exchange, they usually pay in bulk so they could save up on transaction fees. All the excess payments to different addresses that you see, is for other users on that exchange (all of those payments will receive a confirmation at the same time, since it's just a single transaction).
I have heard that larger transactions get verified more quickly because the rewards are more attractive to miners, but if it’s a block of transactions that needs to get verified, then what difference does it make if some of the transactions in it are small?
Correction: Transactions with higher fees (or at least recommended fees), are the ones to receive a confirmation faster (not larger transaction, in case you meant the amount that is being sent).
Also, I assume several other transactions have been verified in the past two days, so why is mine taking so long?
Because of it's low transaction fee (already has confirmations), it moves down in terms of priority.
Aren’t transactions and blocks verified sequentially in time order?
Unfortunately no. Depending on the transaction fee that you pay, you'll be competing against other users transaction (in terms of having it included in the next block that will be mined).