No, on a service change all the wires have to be able to carry the current of the highest rated device.
There are tow cables associated with your question. The one from the pole to the meter and the one from the meter to the main electric panel. The one to the meter is the responsibility of the power company and may well be large enough. Since it is in free air its current carrying capacity is not de-rated in the same way as the cable from the meter to your main panel. Since you will have to likely upgrade your main panel it is a good idea to replace the service entry cable to 100 A rating.
A circuit breaker is intended to protect the wiring of the circuit from overload. The larger a CB is, the faster it will operate, thus in a short circuit situation, the (say) 60Amp breaker will trip before the 30 amp one that is in series. You should only have a moderate factor of safety in your choice of CB. The wiring itself will have a recommended current rating and this is a guide as to the CB.
Rated for exactly 60amps you will find limited breaker spaces due to the rating of the hardware used in it's construction and air circulation restrictions from the equivalent limited enclosure area. When rated at a determined amperage level it is required to construct the device based on the worse case possible scenario which includes a continuous running load. This will determine it's minimum standards according to local and national codes for hardware, room, spacing, and, layout. This does not limit you to only this size panel though. As long as you install the 60amp breaker first in your house panel anything installed down stream {from the load side of the 60amp breaker will be protected up to 60 amps. Therefore you are permitted to exceed those ratings and still remain within code because regardless of what your load is, it can not exceed 60 amps without tripping the breaker. You can install a 12/24 {12-full size or 24-half size [mini's]} 100amp panel for additional breaker spaces and the cost difference is negligible. One code issue to consider is if the sub panel in the garage is not in line of sight of that 60amp breaker then you might want to install another 60amp breaker in the garage panel to interrupt the incoming power giving you the ability to work on/in the garage panel without the possibility of someone re-energizing it mistakenly. I would suggest before this to consult with an electrician for a load calculation to determine if there is 60amps of available power in the 100amp panel in the home. This panel may not have an over-current 100amp main breaker which means it is possible to exceed the panel rating of this unprotected panel. Installing a 100amp over-current device {breaker, fused safety switch, fuse block etc...} like your 60 amp breaker to the garage it will limit the total load to only 100amps. Good Luck and obtain all permits and full understanding of the project and safety protocols in it's regards.
# 6 copper wire.
NO must be 50amp
60amp
Use AWG # 6 wire.
Normally buyers have an inspection that would find a problem like this. The residential disclosure in most states has the seller disclosing any problems. But if the 60 amp wiring is working then it isn't a material fact that must be disclosed. If you had a buyers agent representing you I would be shocked if they didn't tell you to have a home inspection.
in the engine compartment next to the coolant reservoir along with it's 60amp fuse
A #6 copper conductor with an insulation factor of 75 and 90 degrees C is rated at 65 and 75 amps respectively.
2001 honda civic under hood fuse box in slot 18 60amp fuse (EPS) what does this fuse control?
In a 1999 Ford Explorer : The 60 amp maxi fuse in the power distribution box located in the engine compartment is for the instrument panel fuse box
You can. Using these size breakers in a 60 amp sub panel might be pushing the limit for breaking the sub panels feeder breaker. The breaker feeding the sub panel will have to be no bigger than 60 amps because of the main bus bar capacity of the sub panel. It would be better to install a 100 amp sub panel and then there would be a bit of a buffer and you will have the ability to add additional small load circuits.