Is the Universe alive?

Yes, it is.

You can call it God, you can call it Buddha, you can call it Allah, you can call it Universe, you can call it Life, you can call it Truth, you can call it Energy or Light. Whatever name you want to give it, it's not in the slightest bit important. What's important is that this is existence is alive. It is the non material, yet living existence. But it's not something that can be seen or known from within the human mind. If it could, I would simply tell you:

Hey, the Universe is alive and is speaking with you all the time!

But you would never believe it, would you?

How does the Universe communicate with us?

There is a famous book called "Conversations with God" where the author claims to have been conversing freely with God (Universe). However, this is not the way to communicate with God. The reason why this is not completely true is because one creates a separation between oneself and God (Universe) and God is not like that. The author is simply possessed by the entity he calls God (Universe) from within his own mind. Only when you completely discard your illusionary human mind, and to return to this existence within you that is the Body and Mind, the Spirit and Soul, the Mother and Father, and are reborn from that place will you come to know everything with wisdom.

People will see this existence through coincidences, and what Carl Jung famously termed as "synchronicities". There is also the idea that one can manifest according to one's will, with a famous book titled "The Secret" which outlines this, in particular the law of attraction. Again, while one can see that existence through these ways, the fallacy is there is a separation between one's self and that existence - everything is thoroughly one.

The principles of manifestation are true, but the mind that separates oneself from the Universe is not, and so one must have a clear mind

How can I feel the existence of the Universe?

To the extent that your false self is discarded, is to the extent that you can feel that existence. Even in the latest science, scientists are claiming that there is no separation between anything and that separation is an illusion. This is true. Everything is simple that one existence. Everyone is the whole, and the whole is everyone. It is only the human mind which says "this exists" or "that exists".

In religions, they all claim to have that existence and to know that existence. But shouldn't they be in agreeance with one another? Instead, each claims to have the answer and claims that the other religions are all cults. The idea of a cult comes from within the human mind, which separates, divides and categorizes.

In order to truly know, you must completely discard all conceptions of yourself, be reborn as that existence, and from there you will come to know on your own. It is the objective reality outside of the human mind that is undeniable, irrespective of your conceptions and customs.