Insured interests are the lawful interests of the Policy Holder associated with the risk of losses arising out of the Policy Holder's activity due to a change in the [operational] environment in which such activities are carried out [for reasons]/ [caused by circumstances] outside the Policy Holder's control, in particular:
...

I. не противоречащие законодательству - я бы тут через "противное" переводила = lawful (иногда можно встретить reсognised by law - но, на мой взгляд, это чуть Уже смысл передает)

II. объект страхования

Тут, наверное, стоит сначала разобраться с понятиям предмета договора страхования и объекта страхования
(1) предмет договора (= услуга, оказываемая страховщиком страхователю (i.e. несение страхового риска в пределах страховой суммы)
(2) объект страхования (что страхуется) = например, имущественные интересы.

Это по сути Insurable interest = interest IN the life or property covered by the policy

https://www.lloyds.com/Common/Help/Glossary?Letter=I
"Insurable interest - If an insured wishes to enforce a contract of insurance before the Courts he must have an insurable interest in the subject matter of the insurance, which is to say that he stands to benefit from its preservation and will suffer from its loss"

https://www.collinsdictionary.com/dictionary/english/insurab...
insurable interest in British
noun
law
a financial or other interest in the life or property covered by an insurance contract, without which the contract cannot be enforced

insurable interest in Insurance

noun
(Insurance: General)
If a person has an insurable interest in the life or property covered by an insurance contract, they would suffer financial loss if the thing insured were damaged or destroyed.
In property insurance, the beneficiary is required to have an insurable interest in the property insured. You may not take out insurance on an object or a life in which you have no insurable interest. If a person has an insurable interest in the life or property covered by an insurance contract, they would suffer financial loss if the thing insured were damaged or destroyed.

III. по поводу "связанные" и откуда тут "ноги" растут (опять же interest IN)

Определение объектов страхования в законодательстве РФ:
http://www.consultant.ru/document/cons_doc_LAW_1307/3859aa86...
...
4. Объектами страхования имущества могут быть имущественные интересы, связанные с риском утраты (гибели), недостачи или повреждения имущества (страхование имущества).

То есть "связанность" здесь в том, что у лица должен быть фнансовый или иной интерес в сохранении объекта страхования.

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Note added at 17 hrs (2018-01-13 15:31:44 GMT)
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Уточнение: lawful proprietary interests ...