Need to give an IP to our website and I want the HTTP traffic to go though Cloudlflare. How do i get an IP address that will always point to my site? Can I use the ip in the currently published A record? Will it change? If it will change, how often?
Welcome to the Cloudflare Community. 
We cannot answer the question of what IP you should use, at least not with an IP address as that answer. You need to ask that question of the company that operates the server hosting your website. They would also be the only ones able to answer your other questions.
Where is your website hosted?
I host my website, that IP is static. I don’t need a specific IP. I need to know how often Cloudflare will change the Proxy IP addresses that are published in DNS as my A records. I have to give a single IP to something that points to my website but the traffic has to flow though Cloudflare.
That’s not a viable plan. The Cloudflare IP can change at sny time without notice and can even be different in different parts of the world at the same time.
You will need to have that vendor fix their product so that it can use a hostname or find another solution.
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