The verb that best fits all of your sentences and the accompanying adverbs....
"to watch" ....... " to observe closely in order to check on (transitive); to be vigilant, on guard (intransitive)
The administrator (watches, superivses) the employees thoroughly so that they don't chat on the cell phones during working time.
The copyright owner watches attentively so that nobody uses his property without his permission.
The official will (watch, monitor) the execution of the agreement closely. (future tense)
If these are general statements or habitual actions there is no need to use the past tense.
Your sentences are complicated enough to allow for several different solutions.
-The administrator keeps a thorough eye on the employees so they don't chat ........for example.
-The copyright owner is vigilant that nobody use his property without his permission. (with that and subjunctive of - to use)
-The official will see to the execution of the agreement. (see to = ensure by supervision or vigilance)
Администратор следит, чтобы сотрудники не болтали по телефону в рабочее время.
The administrator sees to it that the employees don't chat on the telephone during working hours.
The problem with making an English translation of this sentence is the conjunction чтобы. It is usually followed by a conditional past tense verb in Russian and expresses possibility, desire, demands,requirement, orders,etc.
It can be translated into English with a present tense verb and a conjunction that indicates purpose ( in order that, so, so that, that,etc.) чтобы can also sometimes be translated with a subjunctive verb, an infinitive, or a conditional verb.
In short, your question is a complicated one, as all of your sentences can be expressed in several ways both in English and in Russian.