Sounds like a bad thermostat, they get stuck close and will not let the water flow. Make sure the coolen sytem is clean or clogged. If coolent level is full make sure the fans are coming on, if there connected to the water pump make sure the fan clutch is not warn out. But if it overheats that fast sounds like the thermostat, if after that is replaced it could be a water pump, but start with the cheapest thing first. Thermosat new antifreze and a coolent sytme flush, then go from there.
because the block is cracked or a head is cracked. or if you are lucky it could be just a head gasket. Has it overheated?
The coolant leak is pretty cut and dry - you have a leak somewhere. The lukewarm heat (assuming you're referring to the car's heater) could be for a number of reasons. Insufficient coolant could be one of those. Your coolant could be burnt out. It could be a problem with one of your heater hoses. You could have a worn out heater core.
Could be overheated coolant or just leaked oil burning off your exhaust headers
Engine could overheat, can bust your radiator hoses. If ran for long while overheated you could blow the engine.
Anywhere buddy
Thermostat fail to open. Waterpump is on the deathbed. None, or not enough Anti-freeze in radiator.
The new sensor could be stuck. Or the wiring could be loose, or the coolant could actually be low. Check the coolant level and the sensor wiring first.
Coolant leaking from back? meaning exaust? If yes, then it's a head gasket or cracked head. Will not start because 2 or more sparkplugs are now wet with coolant.
Could have just overheated
there might not be enough coolant inside the radiator, i would check that first. also the heat switch might not open so that the coolant cant make its circle.there is a small possibility that the coolant pump has air around it and it cant regulate the coolant around the engine.beware, never open the radiator tap when the engine is hot.if you overheated the engine ( redline that is) you could have done permanent damage to the engine.
Don't worry about it. If the engine has not been damaged, all you need to do is keep the coolant reservoir full, and run the engine enough to get it hot at which point it will push out a lot of the air. After it cools again, it will pull in the coolant. Keep the coolant level up by checking it every time you get ready to go somewhere. You'll notice when the coolant level no longer drops. Note that if the engine has overheated, it could have been damaged. If that happened, there could be a failed head gasket or cracked head. When either of those happen, combustion gas is forced into the coolant space, forcing coolant out of the engine and causing the engine to overheat rapidly. Unfortunately, if that has happened, the only option is to replace the faulty parts and re-surface the head.
If the engine is cold... it could mean that the radiator cap is too weak to hold and you need to get a new one. If the engine is HOT... it could mean that you have overheated it and you need to add water after the engine has cooled a bit, to prevent getting burned, be careful and use a towel to remove the cap to add coolant.