1) First of all, I will need to explain what is name server. The name server is the location of DNS server from where you can manage all your DNS records like A,MX,CNAME,etc. Domain provider and hosting control panel both provide option to host DNS service for your domain. You have few two options here. If you want to manage DNS records from domain control panel, you will have to use their default name servers. If you want to manage DNS records from hosting control panel (CWP), you will have to use custom name server pointing to your VPS IP address. Moreover, you can manage name server's from domain registrar's control panel only. If you set name server in CWP, it will not have any effect. Why it will not work? That is not our concern here so I will not go into the details here.
2) In order to access CWP using http://cpanel.mydomain.com, you will just need to create DNS A record for 'cpanel' pointing to your VPS IP address. One thing to note here, after pointing the domain name, you will have to add the port number in URL like "http://cpanel.mydomain.com:2030"
3) I never use digital ocean so I don't know this. But what I guess, it could be the VPS hostname. Generally, it is for identification purpose of your VPS.
4) I already have explained this in point-1
5) Yes, you will need to create a hosting account for your domain in CWP in order to be able to host landing page
To avoid name server's confusion, use the default name servers provided by domain registrar and use their DNS service. When you use the DNS service of domain registrar, any DNS configuration on CWP will be ignored, so you can leave it as it is.
Best Answer
What's the problem
You need to wait for the DNS records to propogate.
How long will this take
It'll normally be done in under 48 hours
As you said the propagation checker said the record was wrong, you'll need to wait
Workarounds
Using the hosts file you can specify fixed IPs for host names, or you can override your DNS server to use one that has already propagated.