@Michaela Strotmann
Nginx is not yet running properly, it has to be able to connect to the local server (this would be "good to go" normally, but you will have to check).
One cannot simply remove existing instances of Apache, MySql or PHP, this should simply not be necessary (the install script should detect them or the image should not have them).
The "inbound security rules" that you are talking about, are only a small part of ACL and other configurations that you have to do: all Plesk related ports should be set/configured.
The good news is that you only have to make one set of rules, you can then use the set for every machine (in principle).
Well, Plesk is installed, but not completely, most of the services will not work yet.
By the way, you are using a D1 machine (as I saw from the screenshot) and that is a whopping € 57 euro per month for a SSD based machine, with limited memory (3.5 GB).
There are many important things to note:
a) the D1 v2 machine is cheaper at € 54 euro per month (768 hours) and is much faster! (both the processor and the SSD disk(s) are faster)
Note that changing the D1 to a D1 v2 machine is a matter of seconds, so just change the VM size (a nice exercise, by the way, to get familiar with Azure).
b) € 54 euro per month can get you a little bit more "oompf" (a better server), if you buy a dedicated server somewhere else, but you will not have the fancy and nice Azure environment
c) the D1 has a 60MB/s (from the top of my head, I can be mistaken) and that is enough, even a little bit too much, since the disk IOPS are not that large (500)
Note that, whenever using a D or D v2 series VM, it is recommend to go for the so-called Premium Storage, since normal storage can be somewhat slow and hence a bottleneck.
d) you had RHEL (apparentely), which is not necessary (it only drives the price up) and changed to CentOS.......but you really should consider Ubuntu: it is one of the Linux distributions with an OS image that is provided by the OS vendor. And Canonical does an excellent job supporting Ubuntu on Azure.
e) you have a free trial at Azure, with some credits: monitor expenses thoroughly, since a buck will be charged very fast
f) you should consider to get familiar with Azure Powershell (it enables you to write scripts and automate all common Azure related tasks, including launching a machine)
and all of the above is just the tip of the iceberg.
Naturally, many suggestions and tips can be given for working with Azure, which is a very mature Cloud platform, that keeps on growing in functionality and convenience.
However, it takes some time to get the hang of Azure, so investment some time in "playing around".
If you have questions about Plesk on Azure, just ask (been there, done that many times).
Regards!