You've asked a lot of questions so this is going to be a really long answer.
AFAIC, the first podcast by Russ Mathews is a combination of sloppiness and snake oil. I listened to some of the latter part of it where he put on a one week CHWY bullish put spread (short the $40 put and long the $39 put) as part of an Iron Condor when CHWY is $42.11 and it then dropped to $38. That means a minimum of $2 of intrinsic value and he suggests buying the short $40 put back for $1.35. Not gonna happen.
He also fails to account for the gain on the bearish call spread when he mentions the $400 loss on the 10 put spreads when tallying the P&L but maybe that's just an oversight.
What's most egregious is that he suggests adjusting the challenged put spread by rolling the $40 short put out a week for a 30 cent or so credit. While he could probably achieve that credit, it ends up being a naked put after near week expiration.
IOW, sell the $40 put and buy the $39 put for this week for a net credit of 40 cents per bearish put spread. CHWY drops to $38 and he suggests that you roll the short $40 put out a week for a ~30 cent credit. Great! Looks like break even, eh? But what happens when this week's $39 long put expires? There's no longer any protection and next week's short put is naked.
Stick to tastytrade for educational material.
Now, your questions:
Why would you ever buy the same strike price when rolling out a bull put spread if the price has moved against you?
You would not "buy" the same strike price. It's the short put that is moving against you. If you believe that price is not going to move more against you, you could roll the spread out (horizontally). Down and out would be preferable since you'd gain more time and more price buffer but it may not be logistically available because of the reduced credit and the additional bid/ask spreads that you'll have to pay.
If the price ever falls below the price you sold the put for, you can get assigned even if the expiration date is rolled one week into the future, correct?
With American options, you can get assigned at any time. This is unlikely unless there is no time premium remaining or if there is a pending dividend and it is larger than the time premium remaining in an in-the-money put (dividend arbitrage).
I don’t understand why some people recommend rolling out (say, for example, one week) and set up another spread at the same strike price.
Each possible roll has its own risk/reward potential. The point of rolling out a week is to generate another credit to reduce your potential loss (the stock has already moved against you, forcing the adjustment). The various possibilities must be evaluated.
Are we only concerned where the price is when the options expire? Or do we lose as soon as the price crosses the strike price we sold a put at?
You could begin losing from day one if the stock begins moving against you right away. The more the stock drops, the more you lose. The time to adjust is before the short put becomes deep ITM because then, the credit will be minimal (the tastytrade video mentions this).
Are we not necessarily worried about getting assigned if the price happens to dip below the price we sold the put for? In this case, Does rolling out just mitigate our loss by taking in more credit for writing a farther-term (more valuable) put, if we are assigned?
Rolling out delays the assignment.
Why do people say that rolling over buys more time? I would think that rolling to a lower price would buy more time. Because once the price crosses in the money you loose no matter when the expiration is. Or is that not necessarily true?
As mentioned, short puts are assigned early when the time premium is gone (or the dividend arbitrage). Rolling out increases the time premium, lowering the risk and delaying assignment. Time is measured in days, weeks, months. Rolling to a lower price does not buy more time. It reduces the risk of assignment by increasing the amount of dollars that the stock must drop before you might get assigned.
These might be helful:
Rolling A Short Call Spread
Learn When to Roll a Credit Spread