How to properly set up an internal (local network) web server?

Introduction

Ok guys, this is a tough one, at least for me, and it's a pretty long question that covers some steps about setting up an internal (local network, or LAN) web server.

I've gone through a deep search on the Internet, but I really couldn't find any resources that fully cover this subject from the proper point of view of a software developer/engineer – this is the level of competence I'm looking for in an answer :).

I know this question has been asked several times here on SO, but I couldn't find anything that matches my needs.

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The real question

Some background

From what I know, companies of medium/large size run internal web servers for team collaboration, resource sharing, and other stuff. This servers are for internal use, thus they're not accessible from the Internet. My goal is to have employees nicely access something like suppliers.domain.tld or dashboard.domain.tld and have a beautiful page where they can operate happily (okay, this wasn't necessary, but I hope you got the idea of no troubles on the user experience).

Technology

I think this is not of primary importance, as the general solution should be independent on the specific implementation, but just to give some hints, I'll use Node.js for the backend.

Domain name

I'd like to have a domain name that the user types in the browser to access the website. No IP addresses explicitly written in the URL bar. As in the previous example, something like sub.domain.tld.

HTTPS

I need HTTPS for the communications going on between clients and server. This seems to be a hard topic, as I read in many SO answers that I might need to self sign a certificate to prevent the red warnings from showing up in front of the employees, but this solution has several drawbacks. (Also, I don't want to teach the employees to ignore these warnings.)

Actually, the company has got a registered domain name, thus I might use a subdomain for the internal web server, using the trusted CA certificate of the already existing website. Does this make any sense, or is it feasible? I'm not asking exactly how to do this, but just some general, but still technical, directions on how to accomplish this.

DNS

Do I need to configure an A record on the DNS so that the local hosts are able to access the local web server registered on the subdomain?

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Conclusion

I know this question is kind of vague. My aim is to get a precise overview of the required network architecture and I'm looking for the required steps I need to take in order to make a state-of-the-art solution. Thanks in advance.