I dont think there is currently a way to measure pain objectively. It would need to deal with measuring signals sent to the brain by nerve cells. But I dont think its entirely understood how this works. There is conventionally different types of pain though.

https://www.mydr.com.au/pain/pain-and-how-you-sense-it/#:~:text=When%20we%20feel%20pain%2C%20such,and%20the%20pain%20is%20perceived.

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No, pain is an entirely subjective experience that you can't really objectively measure considering every single person exoeriences it differently. What may be a minor annoyance to one person is excruciating in another.

In medicine, we often will ask how bad something is on a scale of 1 to 10, but thats a ballpark measurement. Most of the time we visually assess for signs of extreme pain. If you walk into someone's room and they're writhing around, or they're sitting completely dead still those are both alarming. If someone with sickle cell disease says something hurts I usually assume they're experiencing a super high level of pain because they hurt every day. So if they tell us they're in pain they're in A LOT of pain. I won't have that mindset in someone who has no morbid health conditions.

Overall just very situational.

Pain is complex experience that doesn't necessarily involve a physical stimulus. It's kind of like asking if we can measure objective happiness.

We do have tests like the ice water paradigm you mentioned but they are limited as people may withdraw their hands at different pain thresholds (eg if one subject is apathetic about the study they might withdraw their hand sooner than another who is invested and is willing to withstand more pain). There have been more recent attempts at measuring pain perception in the brain. Currently there is research into identifying which regions of the brain activate in response to pain and seeing if we can predict the amount of pain someone is experiencing based on their brain activity in those regions. But generally this requires EEG or MRI which can be expensive and time consuming.

Advances in pain assessment: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6979466/

Phantom pain: www.mayoclinic.org/diseases-conditions/phantom-pain/symptoms-causes/syc-20376272

Nope; it's a response to a stimulus with numerous psychosocial aspects.

Choose a stimulus, say, insert a specific size needle a specific distance into a specific part of the body. If you do that with 10 people, and ask them to measure the amount of pain they experience, you will get a range of answers.

More importantly, if you do the same "test" on the same person, say, twice a day for a week, you'll also get a range of answers. How we perceive a stimulus is influenced by many variables which can change day to day, or even hour to hour.

This last part sounds really interesting! While it's maybe not practical, it seems to me like that could be a more comparable measure.

There's no foolproof way to measure experienced pain, since everyone experiences it differently and has different tolerances. However, experiments like the one you describe can give us some insight into how different people react to pain. Additionally, medical professionals often use pain scales (like the 0-10 pain scale) to get an idea of how much pain a patient is in. This in the future can be used to help develop new pain medication or ways to cope with pain. However, when new technology like for example a neural scanner is used to measure pain, we will learn much more about how different people experience pain and what methods work best to treat it.

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As to if there are differences in how physical and mental pain is experienced, there is some evidence to suggest that they are indeed different. For example, people who have suffered from physical pain often report that it is more "intense" than mental pain, while those who have experienced mental pain often say that it is more "distressing" than physical pain. Additionally, mental pain is often more long-lasting than physical pain.

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