Are non Muslims allowed to build, repair or worship in places such as Churches, Temples and Synagogues in Muslim occupied lands, if they are living under Muslims as Dhimmis or have a peace treaty with them?!
The Ahl al-Dhimma are permitted to worship according to their religion, because peace has been made with them on the condition of paying Jizyah and not on the condition of abandoning their religion.
قاتلوا ... حتى يعطوا الجزية
Fight ... until they give the jizyah
فإن هم أبوا فسلهم الجزية فإن هم أجابوك فاقبل منهم وكف عنهم
If they refuse, then ask them to pay Jizyah. If they respond, then accept it from them and refrain from (fighting) them.
Because of the same, they will be permitted to use and maintain any existing Churches or Synagogues. Also because the Quran has attached blame to their destruction and tasked the believers with their defense:
ولولا دفع الله الناس بعضهم ببعض لهدمت صوامع وبيع وصلوات ومساجد يذكر فيها اسم الله كثيرا
And were it not that Allah checks the people, some by means of others, there would have been demolished monasteries, churches, synagogues, and mosques in which the name of Allah is much mentioned.
تضمنت هذه الآية المنع من هدم كنائس أهل الذمة ، وبيعهم ، وبيوت نيرانهم
Included in this verse is the prohibition of demolishing the the churches of the ahl al-Dhimma, and their synagogues and temples
في الآية دليل على أن هذه المواضع المذكورة لا يجوز أن تهدم على من كان له ذمة أو عهد من الكفار
This verse contains evidence that it is not permissible to demolish the mentioned places of worship, when they belong to the Dhimmis or those disbelievers with whom we have a treaty
As for construction of new places of worship, the ruling depends on the place and how it came under Islamic rule.
In a Muslim settlement, or one which was founded by Muslims: They will not be permitted to build new places of worship, as we can not be abetting the propagation of Kufr.
It comes under the general prohibition of the verse:
ولا تعاونوا على الإثم والعدوان
Do not cooperate in sin and aggression.
And is corroborated by the ahadith:
لا تبنى كنيسة في دار الإسلام، ولا يجدد ما خرب منها
A Church\Synagogue shall not be built in the Dar al-Islam, nor will a ruined one be restored
لا إخصاء في الإسلام، ولا بنيان كنيسة
There is no castration in Islam, nor building of a Church\Synagogue
Furthermore, scholars have claimed consensus on this matter.
In a territory which was conquered by military force: The land again has become the property of the Muslims and so it is impermissible to give it for construction of places of worship.
In a territory that was obtained by a treaty whereby the inhabitants keep the land and pay Kharaj: They will be free to build new places of worship as they own the land.
In a territory that was obtained by a treaty whereby the Muslims will own the land and the inhabitants will pay Jizyah: In this case they will not be permitted to build new places of worship, unless they make it a condition in the treaty.
Sure it is permissible. When Islam accords acceptance to the practice of faith of its non-Muslim citizens then it is obviously rational to allow them to build their prayer houses and upkeep of them. Under Muslim rule, their protection would be the responsibility of the Muslim government. Prophet, in war situations, has categorically ordered not to destroy the worshipping places of non-Muslims, not to kill or harm their religious (non-Combatant) men.