How is it possible that transactions can be added to a block currently being mined? It seems counter-intuitive that the current block can be hashed and transactions can be added to it, however that is what this thread indicates.

Can someone help me connect the dots (not necessarily regarding that linked network attack) and understand how something under active computation can have its underlying data (list of transactions) changed?

In other words, if a tx isn't "bound" to a block via hashing, how do they relate to each other?
You're thinking that mining a block is a single long computing process, like computing the trillionth digit of pi or something like that. This is wrong. Each attempt at solving a block takes almost no time at all, and each attempt is independent of the others. Miners make many attempts every second. Each attempt has a very small chance of solving the block. It's a lottery, not a race. Modifying the block between attempts isn't a problem.

Another common misconception related to this is that everyone is working on the same block. This is also false. Every miner works on a different block.