A 100 amp panel should have a conservatively rated busbar and wiring but this says nothing about your particular panel. If you are asking a question relating to safety it needs to be seen by an electrician.
the main breaker
Only if the form factor and specifications are the same. Usually, if it fits and mates with the panel and is the right current and voltage rating, you are okay.
The main breaker will have marking indicating size of electrical service, or if it is a fuse system....the fuses will have current ratings. If it is a subpanel, it might not have a main breaker. In this case you would have to look in the panel that the feed is coming from and find the size of the breaker that is feeding it. In either case you must look for the manufacturer's label that will have the maximum current the panel can handle, make sure the breaker feeding the panel doesn't exceed this rating.
There are two things to think about here. First of all a 200 amp breaker will not physically fit into a 100 amp panel. This is so designed because the panel buss is not designed to conduct 200 amps before the current is cut off. A 100 amp rating on the panel is the maximum amount of current that the manufacturer states, that can be handled safely. Second if the 200 amp breaker is in the main panel then everything downstream from that breaker has to be rated for 200 amp conductivity. The wire size will have to be 3/0 from the breaker to the first over current device in the sub panel which will be the sub panel's main breaker. The sub panel can not be a load center but will have to be a combination panel. I doubt that the 100 amp sub panel's main breaker lugs would be large enough to connect the 3/0 cable.
For a main breaker to trip under these circumstances the rest of the panel is becoming close to its load rating. When the 20 amps breaker trips the rest of the panel is close to or over 30 amps, the 20 amp breaker takes it over the top. What trips a HWT breaker is usually a faulty heating element. It could be the top one or the bottom one.
the main breaker
200 amps
Only if the form factor and specifications are the same. Usually, if it fits and mates with the panel and is the right current and voltage rating, you are okay.
The main breaker will have marking indicating size of electrical service, or if it is a fuse system....the fuses will have current ratings. If it is a subpanel, it might not have a main breaker. In this case you would have to look in the panel that the feed is coming from and find the size of the breaker that is feeding it. In either case you must look for the manufacturer's label that will have the maximum current the panel can handle, make sure the breaker feeding the panel doesn't exceed this rating.
The panel and breaker have to be of the same manufacturer. This way the breaker will fit into the panel. If the panel has a push in bus bar, the breaker must also be the type to accept the bus bar. If the bus bars in the panel are of the bolt in type then the breaker also has to be a bolt in breaker.
There are two things to think about here. First of all a 200 amp breaker will not physically fit into a 100 amp panel. This is so designed because the panel buss is not designed to conduct 200 amps before the current is cut off. A 100 amp rating on the panel is the maximum amount of current that the manufacturer states, that can be handled safely. Second if the 200 amp breaker is in the main panel then everything downstream from that breaker has to be rated for 200 amp conductivity. The wire size will have to be 3/0 from the breaker to the first over current device in the sub panel which will be the sub panel's main breaker. The sub panel can not be a load center but will have to be a combination panel. I doubt that the 100 amp sub panel's main breaker lugs would be large enough to connect the 3/0 cable.
For a main breaker to trip under these circumstances the rest of the panel is becoming close to its load rating. When the 20 amps breaker trips the rest of the panel is close to or over 30 amps, the 20 amp breaker takes it over the top. What trips a HWT breaker is usually a faulty heating element. It could be the top one or the bottom one.
If you mean can you put a single 20 amp breaker in an electric panel, the answer is yes. An electric panel is typically made to handle more breakers with values that add up to more than the rating of the panel on the supposition that you will never draw full load on all breakers. If you do the main breaker would trip. In your case you are under utilizing the panel, but this is not a problem.
It is inserting a breaker into a service panel.
only if listed in the panel board
No, a Cutler Hammer will not work in a GE panel or vice versa.
I doubt if the 125 amp breaker will fit into a 100 amp box. This is due to the rating of the box only being rated at 100 amps. If this exchange could be made then the service conductors feeding the box must be upgraded to take the 125 amperage that the breaker will allow on the conductor.