Do I leave the meat thermometer in the meat the entire cooking time?

My experience with metal probes left in meat for the cooking period, if it is not a short cooking time, can give higher than accurate readings unless the item is large enough to bury a high percentage of the probe and still hit that optimal middle of thickest area without hitting bone. If too shallow, the metal can transfer heat itself to the measuring point so give a higher than true internal temp. This tends to happen more often for me with less dense meat. When this is a concern, I would suggest using the probe initially to get to temp, then moving it to a new location when it reads that you have reached temperature. It this thermal transfer via the metal is what is occurring for you, then the new positions should quickly drop to the true temperature and you will know you need addition time. The 2nd point is usually used for less time, so does not tend to have the same false rise.

All of this is predicated on using a calibrated device of a type of probe designed to allow leaving in as mentioned in answers and comments, i.e. not plastic, not instant read, etc. Those designed to remain in are often the ones with flexible cords to an out of oven base for reading and some wireless models. There are those that the readout is not remote, but those have the disadvantage of needing to open the oven.

I think the common advice about not touching a bone is incorrect - and as a consequence of that advice, inaccurate ideal temperatures have been published.

I think the part of the meat internally touching a bone is always the last part to become done. Think of the cup of undercooked meat in the bone pocket in a T-bone, in lamb chops, in a pork chop; also consider how an undercooked chicken drumstick will often have a red bone or red fluid visible only when cutting through to the bone.

So my advice is actually:

1. as @rpierce stated, first calibrate your thermometer (0-2?C in ice water means you are probably good to go, and a second check of boiling water at 99-100?C to finish off if you want to be completely sure). You also may want to experiment to learn where the temperature reads. I recommend a tip-read thermometer, but a steel probe with a mechanical dimple in the side is often reading at that dimple.

2. Determine, ideally by looking at photographs of sous-vide cooked meats of the same species you are roasting (example http://www.cookingissues.com/wp-content/uploads/2010/02/steaktemperatureswebsmall.jpg for beef) exactly what temperature you want as your minimum

3. Use a thermometer probe with an alarm that is designed to leave in the meat while coooking, (Great example: "Meat Heat" on Amazon https://www.amazon.co.uk/Digital-Thermometer-Stainless-Probe-Perfect-INCLUDED/dp/B012DS2ZY0 ) and touch it through to the bone where you expect the last-to-cook section will be. I.e. on a turkey, I recommend touching the thigh bone from above. I trust this now answers the question in the title; "It depends on your thermometer".

4. Use the lowest oven temperature you need, to bring the roast to the target temperature within the time you have available. If you use a high temperature, you will have a steep temperature gradient (rainbow of cooked on the outside to moister on the inside) whereas if you use a lower temperature, you will have less or no gradient (evenly cooked interior) which you then may wish to sear in some way at some point during the cooking.

If you use a lower temperature and the meat is evenly cooked to the target temperature, you actually DO NOT NEED to rest. "Resting" such perfectly cooked meat will only cool it down.

It depends on the type of thermometer. Typically all metal/glass probes you would leave in, and plastic you certainly would not. However, if your thermometer is telling you internal temp is reached and 10 mins later you find it isn't, then something is wrong: - you are not testing correctly. Make sure that you are inserting to the center of the meat, in the thickest portion, not touching bone - you are aiming for the wrong temperature.

Keep in mind that continuation cooking will occur after you bring the meat out- the meat will typically rise 10 degrees after removing from the oven.

I would not leave a dial-type meat thermometer in the cooking vessel (oven, smoker, etc.). I have two reasons for this.

First, by leaving it in the cooking vessel, you are heating the entire thermometer rather than just the probe tip, which could produce the off results you're seeing.

Second, the thermometer face plate may be of a material that you don't want to heat up beyond certain safe temperatures.

Probes that are meant to be left in the meat will typically be at the end of a longer wire that runs out to a device. Probe stays in the meat, and you read the temperature off the device. I have used these types of probes a lot for smoking brisket and pork butts.

Now, if it is just too convenient to leave the dial-type probe in the meat, and you're not concerned about my second reason, you could calibrate things to your liking. Keep the probe in the meat, and cook until it is actually done to your liking... whatever temperature your probe is reading becomes your new target for that probe.

You could also use a second probe to calibrate against, or calibrate your probe in a pot of boiling water. If it reads correctly, then you know it's something about being in the oven that's throwing it off.