There is a tool you can put in the inlet. When you apply pressure, a leak will show. If it is internal, the level of the fluid will go down.
A bad radiator is a radiator that is leaking or clogged. If you are loosing coolant then you have a leak that should be easy to find if the leak is at the radiator. If it is clogged your engine will overheat.
Find leak, repair/replace cause.
Smoke from the radiator would be do to a leak evaporating off of it. Find the leak repair or replace necessary parts.
Sounds like you got a leak - need repair
if it is leaking from the radiator you can crack an egg and put it in the radiator and it will solve your problem
A pressure test is done to locate a leak if you can't visually see where it is coming from.
It depends on where the leak is and what you mean by patch. If the leak in in the radiator matrix simply putting something over it will not work. There are products which can be added to the coolant as a temporary measure. A radiator can be repaired by brazing or welding but if it has got to the stage where one part is corroded and leaking it is best to change the radiator.
First, you have to FIND the leak. It can be a hose, the water pump, the radiator, head gasket, radiator cap... let me know what you find, and if you can't figure out what to do, we'll talk.
a hose or radiator may be broken. check your hoses and radiator. where is leak coming from?? front, right?? find general area and continue from there
If the radiator was not fitted in properly or the pipes were not attached correctly, the radiator would leak.
Perhaps the radiator itself or lower hose - try pressure testing system to find leak
Find the leak source and repair it. Usually the cause is cooler hoses.