Edit: 60 seconds science has a nugget of science news a day. It is a quick way to highlight a topic and a breakthrough in the area. When I like the info I dig further into the topic.
In my view: where you want to take your career is up to you. Your manager should provide you with the guidance of how to get there.
Going into management is different than continuing on the IC path (at least in most companies). As a principal or architect, your job is to find the technical areas that should be explored (e.g. should Kubernetes be used, or Rust or something else), what are the implications of these technologies as well as how well they will solve the customer problems. Splitting the project and working with the rest of the team to have a sound solution.
As a manager your job will be to deal with people and conflicts that arise due to various misalignments. Motivating people, making sure projects get delivered, taking responsibilities for things that you weren't aware were happening. Setting direction and making sure everyone is informed on what is going on. These are just the highlights, once you get into the position you get to see the different nuances, e.g. how to talk with person A vs person B. Figuring out what motivates them, what upsets them, what are they good at. Observing people and providing them guidance on how to achieve their goals.
It depends on the company and who do you reach out to from that company.
To increase your chances I will advise you to find a manager that is closely related to the area that you want to work on. In the e-mail that you sent to them explain that you see that currently there are no open positions and you would like to be considered when they have an opening. I am a hiring manager and every now and then get a cold e-mail from a candidate. If they are not a fit for my team I search what other positions are open (I work in a big company) and send their CV to that position. Most people in my position will just ignore it as it is extra effort and they don't get paid for it. Thus my recommendation to find someone on the team that you want to join. I think for a startup or relatively small company (<100) it is ok to reach out to anyone from that company but for the bigger companies targeting the team you want to join is important.
Another approach is to find someone at that company that is doing something similar to what you are doing and connect with them over LinkedIn or other means. Ask them if they will be willing to recommend you (most companies have hiring bonuses so they have some incentive of helping you).
Good luck and hope you get a job with one of your preferred companies.