What would happen if PoW cost were drastically reduced or eliminated? Assuming Bitcoin had a different method for selecting winning nodes and controlling block time creation, please describe step-by-step how the Bitcoin network would be vulnerable to various attacks. How would it play out?

Preferred format:


Bad Actor is selected as winning node
Bad Actor makes changes to an Old Block
Bad Actor re-hashes affected block segments, from Old Block to Last Block — (now possible due to low cost)
Bad Actor processes New Block


Please continue...
It think you have your steps of an exploit wrong.  It sounds like you are trying to describe a 51% attack, but are ignoring several factors

First off, step 1 isn't required for steps 2-4 to occur.  A bad actor could make changes to an old block (for example, removing a transaction) and then perform the proof of work on that block and all the blocks on top.  Winning the current block isn't required until all the blocks on top of the edited one are computed and caught up to the current block height (or really chainwork).  So the order of the attack would be step 2, step 3.

Step 4 also isn't really required.  Once the attacker has a longer chain than the rest of the network, she can simply publish their blocks, and other nodes will perform a reorganization.  Whether it's the attacker or someone else that wins the next block doesn't matter.  The edit of a previous block has already occurred, and the attack is complete.

So, in your preferred format:


Bad Actor makes changes to an Old Block
Bad Actor re-hashes affected
block segments, from Old Block to Last Block — (now possible due to
low cost)
Attack complete


The problem with this is that you are assuming that only the attacker has the reduced cost for PoW.  If everyone has access to cheaper hashing, then it doesn't matter that the attacker has access, because she would still have to out-compete the rest of the network combined.  Also, even if the attacker is the only one with access, there's only a limited amount of blocks she can reasonably recompute in a timeframe.  Every 2,016 blocks the difficulty is recalculated based on the amount of time that it took to calculate the previous 2,016 blocks.  If a decent number of those blocks were calculated in secret at this much higher rate, it will increase the difficulty even for blocks calculated in secret.  At some point, the attacker would be making her own job harder.

You may want to read this question to understand better:  What can an attacker with 51% of hash power do?