Can domain have no IP adress (A record) assigned and can subdomains exist?

Unless I am seriously mistaken, I see two problems here.

The first one is trivial. I am writing this answer a few hours after you have posted the question. This means that the new DNS configuration might not yet have propagated through the whole DNS system. Depending on the various TTL entries in the domain's configuration, this may take a day. A typical stage in this process is that some DNS servers already serve new information, while others serve old information.

The second one is your configuration. I cannot see why you have the following entries:

s1.servers.example.com  NS  ns1s1.servers.example.com.
s1.servers.example.com  NS  ns2s1.servers.example.com.
ns1s1.servers.example.com   A   192.0.2.1
ns2s1.servers.example.com   A   192.0.2.2

As far as I have understood your question, s1.servers.example.com is not a subdomain, but just a host. But you are treating it like a subdomain in that you explicitly delegate it to a name server. This is very unusual, and you should probably leave away that four lines. The basic structure should be like this:

• Tell the world which name servers are responsible / authoritative for resolving all names which are in the servers.example.com namespace. You have done this by the first two lines you have posted in your code box.

• Then just add entries (A records) for all your hosts to those name servers' configuration. Unless those name servers replicate each other, you should add the respective entries to both of them. You have done this for s1 by the last line you have posted in your second code box.

There should be nothing more to do. Personally, I would add an additional entry for servers.example.com itself, perhaps just pointing it to s1.servers.example.com, because experience shows that people will try that host name and will be worried if it can't be resolved.

A record is not mandatory in general but it is good to have it there at least to keep hint how to reach it in case of subdomain (you need to know where is ns1.sub.example.com which is handling example.com - you know what was first chicken or egg ? / you can't ask for ns1.sub.example.com in case you don't know about example.com yet :-) ).

There have to be SOA (Start Of zone Authority) records and it is quite common to have at least NS (Name Server) record. Technically NS is needed for delegation related to other subdoamain as "important" NS record for the zone is located in upper zone (in case you ask for NS you are at the end asking directly on the zone, but to reach it you are using NS from upper zone).

For imagination let assume this structure (example.com, example.net and 192.0.2.0/24 is for documentation purpose and the rest is real in DNS system - of course reduced to one of the many records...):

. (e.g. a.root-servers.net. - 198.41.0.4 ):
  . IN SOA ...
  . IN NS a.root-servers.net.
  a.root-servers.net. IN A 198.41.0.4 ( <= this is HINT - how to reach it)
  ...
  com. IN NS a.gtld-servers.net.
  a.gtld-servers.net. IN A 192.5.6.30  ( <= this is HINT - how to reach it)
  a.gtld-servers.net. IN AAAA 2001:503:a83e::2:30 ( <= this is HINT - how to reach it)
  ...

com. (e.g. a.gtld-servers.net. - 192.5.6.30 ) :
  com. IN SOA ...
  com. IN NS a.gtld-servers.net.
  ...
  example.com. IN NS ns1.example.com.
  ns1.example.com. IN A 192.0.2.10  ( <= this is HINT)
  ...

example.com ( ns1.example.com. - 192.0.2.10 ) :
 example.com. IN SOA ...
 example.com. IN NS ns1.example.com.
 ns1.example.com. IN A 192.0.2.10
 ...
 www.example.com. IN A 192.0.2.20
 sub.example.com. IN NS ns1.sub.example.com.
 ns1.sub.example.com. IN A 192.0.2.30 ( <= HINT to where to find NS server)
 ...
 sub2.example.com. IN NS ns1.example.net. ( no HINT needed as it point out to other zone)
 ...

In this example there is consistent information for NS delegation for example.com in the zone directly and in the upper zone.

 example.com. IN NS ns1.example.com.
 ns1.example.com. IN A 192.0.2.10

In case it would differ it is not an issue but for looking for the authoritative server the one from upper zone (com.) will be used but in case of regular recursion the record from zone directly will be returned... So something else than what is technically used can be shown. This may be an issue in case you migrating the zone to other DNS server - until former DNS server is up all is OK and looks good but once it is powered off the zone is not reachable...

The "upper" zone is maintained through the registrar (usually some web form) but the zone directly (your zone) is maintained anywhere you point the NS record (registrar server, third side server or your own DNS server).

To see the record in upper zone you can use this command (in case of example.com so com.'s DNS server):

dig +norec NS example.com @a.gtld-servers.net.

To get the content of the zone example.com you can use (asking e.g. public google DNS server):

dig NS example.com @8.8.8.8

In case the delegation is working in general but not on premises I guess you have locally defined DNS zone for example.com (e.g. to get internal IP instead of public one) so even you have made NS delegation for subdomain your local instance of DNS zone example.com have no idea about that - you need to add it also there.

Next to it you don't need extra NS delegation for each subdomain. In case you don't need extra subdomain DNS server even "main" DNS server can hanle it.

This is valid content of the zone for example.com

example.com IN SOA ...
example.com. IN NS ns1.example.com.
ns1.example.com. IN A 192.0.2.10
...
www.example.com. IN A 192.0.2.20
s1.servers.example.com. IN A 192.0.2.30
s2.servers.example.com. IN A 192.0.2.40
s3.servers.example.com. IN A 192.0.2.50
...

In case you need delegation just servers "level" may be enough:

example.com :
 example.com IN SOA ...
 example.com. IN NS ns1.example.com.
 ns1.example.com. IN A 192.0.2.10
 ...
 www.example.com. IN A 192.0.2.20
 servers.example.com. IN NS ns1.servers.example.com.
 ns1.servers.example.com. IN A 192.0.2.15
 ...

servers.example.com ( ns1.servers.example.com. - 192.0.2.15 ) :
 servers.example.com. IN SOA ...
 servers.example.com. IN NS ns1.servers.example.com.
 ns1.servers.example.com. IN A 192.0.2.15
 ...
 s1.servers.example.com. IN A 192.0.2.30
 s2.servers.example.com. IN A 192.0.2.40
 s3.servers.example.com. IN A 192.0.2.50
 ...

So in case you don't need extra DNS server for some special purpose directly on s1.servers.example.com this record from the question is not really needed:

s1.servers.example.com  NS  ns1s1.servers.example.com.

You configuration is correct, except the:

s1.servers.example.com  A   192.0.2.1

Since you delegated the domain s1.servers.example.com to other name servers, its their job to define this entry. An exception constitute the glue records, which must be in the parent zone. You can use them to give a more rational name to the delegated nameservers:

s1.servers.example.com  NS  ns1.s1.servers.example.com.
s1.servers.example.com  NS  ns2.s1.servers.example.com.
ns1.s1.servers.example.com   A   192.0.2.1
ns2.s1.servers.example.com   A   192.0.2.2