1) can it be bad if the battery runs out of power and the Pi dies? I know sometimes you need to worry about corruption and stuff.

Yes. The storage medium the OS is stored on may corrupt. The only reason this doesn't happen to other battery powered devices is because they will safe shutdown before the battery dies. "0%" battery is a lie.

2) It would be ideal to power it via the pins (rather than using the USB input) for my use case. From what I understand, I should be able to just input 5V on pins 4 and 6, right? does the USB offer anything (e.g., voltage protection, smoothing, etc) than the pins don't?

The Pi Zero specifically does not. Unless you count preventing someone from plugging it in backwards?

3) is there anything I need to be careful of with this? I.e., I'm powering it via a buck/boost converter like this one that can take > 3V. I'm using a 3.7 LiPO battery cell and tuned the pot on the board to output (measured) 5.1V. I ask because I think I fried a Pi zero W doing this, and I'm not sure how.

The pi is fairly sensitive to power fluctuations (compared to items designed for embedded operation, such as an Arduino). I'm not an electrical engineer, so my knowledge here is limited, but I might consider going with a more expensive, well-known brand like SparkFun for my power needs.

Q4 is way outside my knowledge, so I can't make any recommendations.

I want to power my Pi zero W with batteries, for relatively short periods of time (< 1 hour); I don't really care if it runs out of power and shuts down. I had a couple questions though, if anyone can help me out:

1) can it be bad if the battery runs out of power and the Pi dies? I know sometimes you need to worry about corruption and stuff.

2) It would be ideal to power it via the pins (rather than using the USB input) for my use case. From what I understand, I should be able to just input 5V on pins 4 and 6, right? does the USB offer anything (e.g., voltage protection, smoothing, etc) than the pins don't?

3) is there anything I need to be careful of with this? I.e., I'm powering it via a buck/boost converter like this one that can take > 3V. I'm using a 3.7 LiPO battery cell and tuned the pot on the board to output (measured) 5.1V. I ask because I think I fried a Pi zero W doing this, and I'm not sure how.

4) I need the power system to be very portable and pretty lightweight, hence the single LiPO cell and small board, and not a huge brick of AA's or something. But is there a better/safer option that's lightweight? maybe 18650 batteries or something?

thanks for any advice!