I think that there are two different processes at work.
For the wood stove it is probably the differential expansion of the various parts of the metal frame whilst the stove is heating up which causes the pings. One metal piece sliding relative to another etc causing a vibration which passes through the metal structure which in turn excites the air around it and causes the sound which you hear. Probably the rate of pinging increase because more of the stove is heating up. Eventually when the various parts of the stove reach a steady temperature there is no more expansion of the stove and the noise stops.
For the heated pan with water in it the process is different.
When a pan of water is heated the first bubbles that are seen are bubbles of air formed by the air which is coming out of solution. This is because the solubility of air in water decreases as the temperature of the water increases. You do not hear the result of the formation of bubbles of air.
Eventually boiling occurs and this is when bubbles of water vapour form inside the liquid.
For this to happen a number of liquid molecules have to go into the vapour phase and also a surface has to be created between the liquid and the vapour.
Both these processes require energy.
The formation of vapour bubbles is a complex process but it is helped by the vapour bubbles forming on nucleation centres. In the case of a pan of water these may be imperfections (edges, crevices) in the walls of the pan. If the pan is clean and smooth there are fewer of these nucleation centres.
If there are not enough of these nucleation centres then the liquid becomes locally superheated and the temperature of the liquid rises above its boiling point. Because of the liquid being in a metastable state when eventually a vapour bubble does form, it expands extremely rapidly and in doing so sends out a pressure shock wave throughout the liquid and ultimately the pan containing the liquid. This makes the pan and the stove vibrate and as the pan and the stove have a large surface area in contact with the air this also makes the air vibrate to a degree such that the sound waves can be heard.
These are the pings that you hear and they become more frequent because as time goes on more of the heat supplied to the pan from the stove is used to produce water vapour rather than just raise the temperature of the pan. The greater the rate of energy supply from the stove to the pan the larger the local super heating and the rate of formation of vapour bubbles, the greater the rate at which you hear the pings.
The pings from the stove heating up and the water boiling sound similar because they emanate from the same soundbox - the wood stove.
For very clean and flat pans the effect is much more noticeable. Chemists when distilling, particularly at low pressure introduce “anti-bumping” granules into their flasks to reduce the occurrence of very rapid vapour bubble formation. These small granules have sharp edges on which vapour bubbles can form more easily. Heating water in a microwave oven can result in the water actually being forced out of the container. This is because it is the water at the centre of the container which is hottest and there are fewer nucleation centre in that region so the superheated water can be at a very high temperature. The very rapid and large expansion of a vapour bubble causing liquid to be forced out of the container.