I want to know why there are so many pairs available to trade on exchanges.
It's for you to have enough available currencies pairs that you can exchange your crypto coins with, mostly, you will discover a well recognized crypto coin paired with a newly introduced currency fast rising, this helps other people to invest in them through exchanges from one coin to another and then when their market rises, they developed high market cap and volume and the value of such coin increases, so this help in exchanging from one coin to another for investment profitability on coins that have good and realistic project developers.
Does trading with a bitcoin pair help us save fees in any way?
It all depends on the coin you're trading for, the network they belongs all determines the rate of what you're going to be charged.
How will I profit if the prices of altcoins match those of bitcoin? When I trade with the BTC pair, both will rise and fall simultaneously.
If you're trading, you will see the rate each currency you're making pairs with worth with the standard rate of bitcoin in respect to USD, bitcoin may be rising while your atcoins is falling, except on rare occasions you discover them rising both at the same time.