As to whether you can leave a meat thermometer in the meat as it cooks depends on a couple of factors…
Only certain meat thermometers are designed for use on a closed oven for example. Unless the user manual of your meat thermometer states that it can be used in an oven, then you should assume that it can’t be left in a closed hot oven.
Besides, most meat thermometers these days are known as “instant reading thermometers”. And what this means is that they are designed to quickly probe your meat, and give you an almost immediate temperature reading, allowing you to set the thermometer aside afterwards until you decide to check the temperature again.