I asked the exact same questions a couple weeks back. Weird.
Prayer to ancestors:
How I understand it is in most cultures, there is some reverence of ancestors. America we have tomb tombs, family meals, and anniversaries. Many people in the Philippines honor their ancestors with food offerings (as done in some African traditions). I assume Indian countries do this too as with Japan (where Nichiren Diashonin, a Buddhist monk, said never to forget who bore you).
I notice these customs and prayer are incorporated in that person's or family's faith. For example, my friend who is from the Philippines is Roman Catholic. They do not teach to pray to the ancestors; however, in her country giving offerings to passed on birthdays and anniversaries are very common. Likewise in Indian cultures and I'm sure other Asian cultures do so as well.
So, long story short, I don't see them conflicting. I pray and honor my ancestors and I am relearning the Buddha's teachings so I can follow them. It's, ancestor veneration, more of a cultural practice or custom rather than part of the belief. I don't know about Theravada, though.
Do Buddhist believe in praying?
Yes, very strongly. I can say safely all Buddhist sects believe in prayer. Most refer to it as chanting and mantras. The meaning is pretty much the same. Asking for blessings, honoring, worship, and/or whatever means of communication with oneself Buddha-nature, the Buddha, various other Buddhas that mirror The Buddha Himself
and/or the like.
Does the ancestors and old Buddhas live in a spirit world, where they can communicate with living people?
I have to re-read the Sutras. I read (outside the Sutras) that Buddhas believe in reincarnation; but, it such a common stereotype associating Buddhism with reincarnation that I don't know if it is true. I was told it came from the Hindu; they believe in reincarnation.
No, I don't remember Buddha teaching about the spirit world. Rather, He was opposed to a lot of that type of thing "deities, demons, and so forth" of His day. If anything, our ancestors are still on this earth in spirit. Though I don't know how that relates to Buddhism well.
Also, I know the Buddha taught in metaphors when explaining about the human condition by using "spirits, dragon kings, demons, and heavenly angels" and so forth. I don't know if it is actually a metaphor or if there are actually these beings within the Buddhist teachings.
Why does some buddhistic traditions pray to theire ancestors or older buddhas?
Does buddhists believe in praying? does they believe in a life after death?
Does the ancestors and old buddhas live in a spirit world, where they can communicate with living people?
Some resources:
The Basics of Buddhism
Some are good guides. I rather depend on the Sutras; but, there are so many.
A Basic Buddhism Guide: 5 Minute Introduction
Vinaya Texts, Part I (SBE 13) - Mahâvagga - First Khandaka Four Noble Truths
The Buddha's Teachings
Afterlife and Buddhism
afterlife in mahayana? | ReligiousForums.com
Other
A Complete Buddhist Sutra Collection or
Online Buddhist Sutras, Scriptures, Texts and Related Media, in Chinese, Sanskrit and English, for Free
Buddhist Sutra - Sigalovada Sutta (Lay person's guide "Sigalovada Sutta") I haven't read the Sutras in a while. Most of them I read so far are more for the Monks (how high beds should be, what to wear).
Nichiren Diashonin also talks about the ancestors and honoring the deceased. In the Lotus Sutra there are many Buddhas that The Buddha said will come after Him. So, these are good to keep in mind too. Mind you, He is from Japan and takes from His culture and Ti-en Tai sect point of view.
The Origin of the Service for Deceased Ancestors | WND I | Nichiren Buddhism Library The Origin of the Service for Deceased Ancestors
On Offerings for Deceased Ancestors | WND I | Nichiren Buddhism Library On Offerings for Deceased Ancestors