Okay, I'm finally ready to answer this. I apologize in advance for how wordy this is. I got excited in my spiral of research and ramblings. If you read this (and the attached notes in further detail), well, then you are a saint. haha. 
Pain is multifaceted. It is both experiential and emotional. Pain is a natural defense mechanism which warns the body about potential injuries, threats, diseases. It alerts the individual that protective actions must be taken. 'Fight or Flight' response is also common.
There are physiological processes which occur when we experience pain; specifically, 2 distinct pathways in processing via the spinal cord to the brain.
This process is also psychological considering the psychosomatic aspects. The latter is just now being researched in studies by Dartmouth, U. of Michigan, U. of Melbourne, U. of Colorado, etc. -- as there has been a neurological pathway overlap discovered in how emotional pain is processed similar to physical pain. Furthermore, there is work that was conducted by the late Dr. John Sarno, regarding experience of emotional pain, and how it corresponds to chronic pain disorders in the body. A lot of his work found that external stress or anxiety can manifest as referred pain in other parts of the body as the brain tries to make sense of it. Wherein someone experiences unprocessed past traumas as pain or tension in their lower back, or tightness in their chest, for example.
I will outline the physiological processes first, as I find biology highly intriguing, and love the way the body conducts a series of fail-safes just in case we need to protect ourselves. haha. I will explain the psychosomatic correlations after, and then answer your last question as a hypothetical, once I've gone through how we actually experience physical/emotional pain and what the body does to respond. For the sake of not being too verbose, I will do my best to explain these pathways as concisely as possible. However, I will add my additional notes in an attachment for the more technical aspects of these processes, pain phenomena, and further research, should anyone care to read them.
First, imagine you are stabbed in the abdomen by a sharp object. There are a number of things your body will do quickly in response (there are 2 pathways Ascending and Descending to process the pain information and regulate it).
3 types of pain:
Somatic Pain: Pain from injury to the skin, muscles, or joints.
Visceral Pain: Pain from internal organs. Often experienced in a different location- a phenomenon known as ‘referred pain’. This is due to the convergence of the pathways in the spinal cord at certain levels. The brain does its best to make an assumption of location based on the number of receptors which are stimulated. (Ex: Heart attack -> felt in shoulder, left arm, or back instead of in the chest where the heart is actually located.)
Psychosomatic Pain: Pain from emotional or mental states. This pain is processed similarly to physical pain, and will often manifest in physiological forms (new studies suggest an overlap of brain processing).
How the brain processes pain:
Nociception- a biological process, and part of the nervous system’s protective response to harmful or potentially harmful stimuli. This response travels two distinct pathways.
-Ascending Pathway: Responsible for transmitting the pain signal up to the brain for processing. Sensors in specialized nerve endings detect mechanical, chemical, or thermal threats. If enough sensors are activated, electrical signals ascend up the nerve to the spine, via the spinothalamic tract and into the Thalamus, then carried further into other regions of the brain. The brain ‘weighs’ the importance of these signals, localizes the source of pain, and produces pain sensation if it decides the body needs protection.
-Descending Pathway: Responsible for controlling and inhibiting the ascending pathway. The brain sends signals to the brainstem, which stimulates the production of serotonin and adrenaline as inhibitors to the electrical signals and chemical processes (limiting production of the chemicals released in cell-inflammation or death). It also signals an interneuron to create endogenous opioids to inhibit the processing of the information of the pain, and to prohibit the synapses of the neurons within the spinal cord. Thus, the pain information is lessened, and the brain is able to process more clearly how to proceed.
Psychosomatic Pain: (often referred to as ‘mind-body connection’)
-Pain which is experienced first in the mind, and then in the body. Physical pain has sensory, cognitive, emotional, and social elements, rather than a unitary phenomenon analogous to nociception. Emotional and physical pain are linked in brain processing.
-Due to the social aspect of human behaviors and experiences, and given that we are social creatures, when we face emotional traumas, social rejection, heartbreak, loss, loneliness, or fear; often, the processing of these emotions psychologically can lead to physiological pain, referred pain, or chronic pain (pain lasting more than 3 months).
-Often, there is a psychological cause when there are no physical reasons for the experiencing of pain. This is evident in those with anxiety, stress, or recall of trauma. The body reacts physiologically, increasing heart rate, constricting blood flow (muscle aches, headaches), etc.
Would we experience pain in the same way if we didn't have a mind?
There are several ways I can answer this given the context of the question. If you mean brain when you say mind, then no.
However, If you mean cognitive processing when you say mind, then I'd say if we didn't have cognitive processing, we'd experience pain only in the physiological processing of it-- likely mechanical as a means of survival (the 2 pathways might exist, but the brain would lack capability to process, adapt, or regulate). Regarding psychological pain, if we didn't have a 'mind', then we'd likely be incapable of being affected by social-emotional pain or traumas. We'd experience them, but have no means to process them, to create defense mechanisms (other than opioid production), or to work through them.
Now, if you mean "would we all individually experience pain in the same way", then I'd still say no. The brain is adept at receiving pain information. We'd still have people who get pricked with a needle who jump 5 feet in agony, and those who barely flinch. The physiological processes would be similar, but perhaps the adaptations, defense mechanisms, or regulation might not be possible without a mind. Given the studies on pain and personality, they indicate that pain is subjective due to numerous factors which influence it (I've added a section about that in my notes attached).
What do you think, Ren?
