Please Help, I have spent a month going around in circles trying to figure this out.

I wish to design a site which from a standard unregistered visitor looks much like a company/employee director, with basic company information including logos, and employee details (all made as pretty as possible).

However, when a company joins the site (at this point that would be for free, but later I would like to possibility to sell subscriptions) then the company administrator would add the company details as well as set up employee accounts, with each employee having their own login (Might need a custom login fields for example: company name, username, and password)

Look at this module: https://www.drupal.org/project/group

This is an alternative to Organic Groups (but it's production ready for D8, and trust me, you want to use Drupal 8).

So there is an element of company/employee directory, but the other side of the site is employees from a registered company can login in to create a client database, and each client can have an unlimited amount of prebuilt forms associated to him/her, with some forms having to be associated to a client more than once. For example, if you have an address form, that is completed for every client, and that client moves then the new address cannot over write the old address, both addresses will need to be kept and accessible by the employee.

This is really just a bunch of entity types, probably nodes, with various fields - as VM says you can use multi-value fields and use Views or other methods to only output the most recent record, but the old records stay in the database. You can get a lot more sophisticated if you use Paragraphs module as well.

(for now) Clients will not have a client login and all client information is private and cannot be seen by anyone else (maybe seen by the main company login, not sure yet).

OK, but if a client could login, what could they do? You need to think about that now.

That's a start - Groups, maybe Paragraphs, and Drupal core (Views, Fields, Content types etc). Of course you'll want a lot more stuff and the real learning curve will be in Drupal core - getting down with fields, content types, security, etc etc, and then you have to theme it.

Like I said, Drupal 8 eats stuff like this for breakfast - it's fieldable entities + entity reference system is all powerful and no other CMS has anything even like it, and then you have Views (a list builder) and it's granular permissions system etc etc.