Both of these are examples of normal, straightforward English - neither is idiomatic or unusual at all. 1. The verb form 'is found ' is just the passive of the verb 'find'. For example, if experts examined an artwork and discovered that it was forgery, you would say 'The painting was found to be a forgery'. This isn't an idiom. In fact, it is a common passive construction. For example, if people think that someone is a hero, you could say that 'He is thought to be a hero'. 2. If you want to refer to the area of sea a short distance from a coast, you need to use the preposition 'off'. For example: 'The island is off the west coast of Scotland" or "The ship sank off the coast of Portugal" , and so on. You could also add the distance - for example 'The collision occurred three miles off the coast of Florida'. It is very clear from the context what this means. This is a common and standard use of 'off', but for some reason, students rarely know it. It's definitely worth remembering.

1) "Found to be" is a standard expression, if maybe a bit formal. It generally means to uncover a fact about something, or develop an opinion about something. For example: a) The doctor found the patient's leg to be broken = the doctor determined the leg was broken b) I found that movie (to be) boring = I think the movie was boring c) The jury found him (to be) guilty = The jury determined he was guilty 2) I would say generally "off the coast" indicates that the location is nearby, but it could also be used for something more distant. To me, what matters is that there is nothing between the coast and the location. Hawaii is "off the west coast" of the US, even though it is far. If you want to emphasize that it is close by you can say "just off the coast".

Found to be means to be discovered or realised, if you like. I found the shirt to be poor quality, i didn't find the film very funny. In the second question, "off" is in reference to location. I would asume from just saying "off" alone it would mean closer but you can say "just off" which would definetly mean close to and if it's far it would be said like "the island are about 500km off the coast"

The phrase "found to be broken" just means that the object is discovered to be broken and this fact is slightly unexpected. In other words, the person looking at the object discovers that the object is broken. I wouldn't consider this an idiom. "Off" in the second sentence means in the water near that edge of the country. When we say "off the coast of" it always means in the water by the coast.