If you mix the old green antifreeze with the new type, will it gum up and cause the car to overheat

Antifreeze is just not another fluid to spend money on, it is important to replace this vital fluid every two years and here's why. Antifreeze coats the water pump and the thermostat and other engine metals so that it does not rust. Without it the thermostat would stick and cause the engine to overheat and destroy the engine, it also lubricates the water pump bearing making the pump last much longer without. If you run a car without anti freeze it will destroy your water pump and thermostat and pump it will also allow the engine to rust on the inside as well. Antifreeze also keeps the water from depositing harmful chemicals inside the block for instance if the water is high in calcium the heat coat the steel with the calcium in the engine will cause the engine not to cool it will cause a coating of calcium on the metal parts that will cause the antifreeze not to be able to draw the heat away from the metal causing hot spots in the block and causing premature engine failure. Always Mix the antifreeze with 50% water and %50 antifreeze this gives you the best protection for high temperatures and for the lowest temperatures. IF you change the antifreeze yourself be sure to catch the antifreeze in a bucket and dispose of properly. Antifreeze will attract animals cats dogs and birds and they will drink it and die so do not leave it out where pets or children can get to it. bottle it up and recycle it.
Many cars have different antifreeze types for instance Buick and other GM products has a gel like antifreeze and it needs to be change with the same type or it will damage the engine. many others require glycol antifreeze and this can be bought at any parts store. When changing antifreeze it is always a good ideal to replace the thermostat and the radiator cap. The reason is you already have the fluid drained and it is best to do it at that time it saves labor and time. Radiator caps will wear out, the rubber gasket fails over time and will not hold the correct pressure on the cooling system causing the engine to overheat, since the cap keeps keeps pressure on the cooling system it keeps the water or antifreeze from boiling over. Once the cap fails it will allow the liquid to boil and it will overheat the engine, What you can't see is where the liquid does not touch and that's next to the cylinder walls there air bubbles can form and cause hot spots on the cylinder walls, this causes ring failure and reduces the life of the engine dramatically.
Just a thought for you that love animals, they do make antifreeze that is pet friendly, (pets won't Drink it) most antifreeze is made out of crushed orange peel and the oil in the peel is sweet to animals and they are addicted to it, however this will kill them it shuts down their stomach. Just in case you want the pet friendly type anti freeze. Also if yo spill antifreeze you can buy a bag of cat litter to pick it up with. To do this spay a small amount of water on it then use the cat litter to pick it up put in a plastic bag and dispose in a trash can. The more you dilute it the less it attracts the animals.
While you are doing this now is a good time to check that the overflow tank hose does not have a hole in it or the overflow tank if either has a hole in it it can pump water out of the radiator and cause you to lose fluid out of the cooling system and cause the engine to over heat.