It's open source, however, if you want your solution to be deployed on their cloud or on the cloud in general without you being involved and having it and managed by them, there may be costs involved. That's the paid feature.

Jet Brains allows using their product at no charge, but with a cap on build agents, and the number of build configurations allowed, so you can explore the capabilities before committing.  Our current licensing level allows up to 10 simultaneous build agents running, and unlimited build configurations for less than $2000 per year; although not free, it seems a very reasonable cost for the horsepower delivered.

The licensing costs depend on what you use the solution for, however, it's free to start and you get up to three agents for free. If you want to do more than a hundred builds or a hundred different setups, then you have to pay more or start paying. 

It's a Freemium model. Once you pass the free stage, it can cost anywhere from a few hundred to a few thousand dollars per year.

TeamCity is on the expensive side. It's more for developers than CIS admins. Conversely, Ansible is more for CIS admins and less for developers. It would be nice to have a solution that works for both purposes. So I think Ansible was something they were thinking about purchasing, but I'm not sure if that ever occurred.

Start with the free license it has decent room for one group.

The licensing is on an annual basis.

I cannot comment on the pricing, as this is out of my purview. 

The initial cost was only the cost of a virtual server, and database which was less than 1000 Euros a month. I am not aware of ongoing costs.

Start with the free version, and go from there.

Start with the free tier for a few build configs and see how it works for you, then according to your scale find the enterprise license which fits you the most.

Setup is the licence (and my contract rate).

I don't handle pricing so I can't really advise on this.

If you can, try it out first.