There are a few free and very good services available online where you can check the state of your IPv4 IP address against multiple blacklists.
Blacklist Checking Services
With the above mentioned services you can check your IP address state against many blacklists for mail spam, open proxy listing and etc.
Usually to get your IP address delisted you have to verify that whatever has caused the listing is NOT happening anymore. If that is checked you have to visit the actual website of the blacklist and look for a "Delisting" or "Removal" option. There you can submit a request for your IP address to be removed from their blacklist. This usually works well.
Example of a delisting or removal request site of a blacklist:
http://www.barracudacentral.org/rbl/removal-request (Barracuda Blacklist)
Others can be quite different. Sometimes you might have to sign up with them and open internal tickets for removal of your IP from their blacklists and others only accept mails.
About your IP address "receiving very heavy bot traffic trying to access the IP". This is a very common problem and absolutely nothing new. You can change your IP address as much as you want but it will still keep happening (sooner or later). The bots scan all subnets on the Internet and attack all IPs. There is no escape. You however, given that we discussed this before here, know what to do to get rid of the automated attacks
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. So the best weapon against these attackers is a secured server. This is always a rather good idea. Operating a server wide open and without any kind of security is simply negligent and shows a very modest behavior.
Generally IP blacklisting is also a very common issue in the hosting world. ISPs and providers should care about it. There are extreme cases where blacklists list whole IP ranges that are probably only rented from datacenters. And the datacenters terminate contracts and service with the people who rented the IP ranges from them because they care to have a good reputation. Especially for servers a blacklisted IP means a lot... atleast if you're ever planning to host a mail server or pass traffic through the IP (e.g. coperate VPN, private VPN and such).
Imagine you are running a hosting company and someone abuses your IP space by spamming thousands of mail inboxes with a overall of millions of spam mails. This would get you listed on pretty much all of the blacklists. In the worst case you get the whole IP space listed and the datacenter where you got the hardware and IP space from won't be happy. They will decide to cancel the contract with you. Out of the sudden the plug in your IP space is pulled. Clients demand their money back, start PayPal and CC disputes and run away. Farewell "your hosting company name here".
Users themselves should also care of course. When I got VPSs before I performed blacklists checks and made sure I don't run anything that might cause the issues. Then I went to the blacklist sites and requested delisting. In most cases successfully. Some blacklists can be really hard to get off because they only accept removal requests from the owners of the IP blocks or won't remove a single IP address on request if they have listed the whole range.