Some won't boot without a detectable battery cell. Depends entirely on the laptop in question what the best course of action is. Most newer bios handle charge profiles automatically and will prevent ac related damage but it's all dependant on how they were designed/made.

It does not matter if the battery is plugged in or not. Far more important is the state of the battery. All LiPo batteries degrade over time. But they can degrade faster or slower depending on the state they are stored in. They degrade faster when at higher charge levels or when stored in hotter environments or if they go through more charge/discharge cycles. Older battery technology also degraded faster in general, new ones tend to last longer in sub-optimal conditions.

Apart from newer battery technology itself battery monitoring and charging technology has also improved. A lot of modern laptops have smarter charging circuitry that lets them stop charging before the battery is at 100%, sometimes configurable in the bios, sometimes controllable via the OS. This can help a lot to preserve the battery life for longer, especially if you leave it plugged in as it spends less time at 100% charge. Older devices also tended to run hotter for longer periods of time, even when idle. Both of these combined with worst battery technology would lead to batteries degrading quite a lot faster if you left them plugged in all the time - hence where the advice came from (note that removing the battery at 100% charge was also not great for it, better to store lipo batteries at 40-60% charge, but it did still save it from the heat of the device) . But when setup correctly modern devices suffer from this a lot less so it is much less important to remove the battery at all - I doubt you would really notice the difference overall on modern systems.

Removing the battery when using AC used to be the advice to prolong the battery life a decade ago.