What do Hindu scriptures say about Hindu temples?
General perspective on temple worship: Deity worship was the main process in dvapara yuga. In kaliyuga the recommended process is chanting of the holy names of Lord Hari. You can see this for more information on this point.
Do Hindu scriptures say that we should build temples? (Yes | No)
Building temples is beneficial for our own self and for the society. Couple of scriptural citations:
One who builds or helps to build a beautiful temple for the Lord will be freed from all sinful reactions and will enter the Vaikuntha planets. - [Narasimha Purana]
One who offers the Deity gifts of land, markets, cities and villages so that the regular daily worship and special festivals of the Diety may go on will continually achieve opulence equal to My Own - [Srimad Bhagavatam 11.27.51]
If I go to Lord Shiva's and Lord Hanuman's temple, but I also have the deities at home temple (puja mandir), can I ignore going to the public temple? Is it necessary to visit a (public) temple which has "Pran Pratishtha" instead of the normal temple at home where we pray and worship God but God goes away after worshiping? (Yes | No)
Generally a temple where the deities are worshiped according to scriptural injunctions is more conducive for spiritual advancement. The atmosphere is clean and spiritual because the sole purpose of the place is to facilitate deity worship and other spiritual activities. Having said that in an advanced stage when our heart is filled with devotion we will be able to feel that sanctity all over and our worship-able Lord everywhere. However, this is not the position of normal people but highly elevated. So unless the temple is far away it is better to go to a temple regularly. If the temple is far, one can worship at home daily and visit temple on a regular basis.
Are there any rules for home and public temples? If yes, what are they?
I can only answer from Vaishnava perspective (Gaudiya Sampradaya) for worshiping Vishnu deities. This site, under 'Worship' menu has Temple and Home Worship sub sections. Under them you will see other sub sections.
Is it necessary for a Hindu to visit a temple at least once a day? (Yes | No)
Vedic scriptures don't make anything mandatory. They recommend the best things for us and it's up to us to seek and follow. If you do go to a temple, it elevates your consciousness and brings you closer to attaining self realization. If it is impractical to go daily make it a weekly or other suitable periodical visit. Also when you take up a spiritual path, it is your guru who can give you the best advice.
Did Hindu scriptures originate temples or is it the people (Brahmans/devotees)? (Scripture | People)
There are ample references to temple and deity worship in Vedic scriptures. For one such instance read chapter 11.27 of Srimad Bhagavatam: Lord Krishna's Instructions on the Process of Deity Worship
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