Was there an equivalent to "start-ups" in socialist countries? How would an employee/factory director who had an idea for new product or method for producing go about it? Did they submit applications to State planners or ministries and State funded it?

There were no start-ups in the Soviet Union. Trotski and Bukharin rejected the idea of a labour market, substituting it with a militarization of labor. This made the state the only recognized legal employer. There were some limited reforms over the decades, but the basic template remained, meaning you didn't just start your own business to compete against other companies. By the 1980s there were only 2 main retail networks, 1 airline company, 1 oil-extraction ministerial monopoly, and 9 ministerial concerns in defense industries. A not super rigorous but still useful introduction is this 1995 paper.

In reality, rather than speak of entrepreneurs and startups, you had intrapreneurs, working within the strict constraints of a universal state monopoly, potentially creating not new businesses, but new administrative units. These could be party figures, scientists, and even Gulag R&D labour camps. The latter, called sharahka, were R&D prisons, where scientific convicts were tasked with designing innovations for the state. They were spared hard labour, and successful researchers could even receive medals from the state. This is a good basic intro: www.rbth.com/history/332537-how-convicts-designed-soviet-weapon/amp. This is the most extreme illustration of the dynamics of intrapreneurship and innovation in the Soviet Union. You could not be an entrepreneur or create an independent start-up, but you could be an intrapreneur innovating in the framework of the political, military and economic interests of the state/Party, and earn, not money primarily, but political/administrative autonomy and scope, even in prison, which were the real currencies in the Soviet economy.

If an innovator/intrapreneur had an idea, yes they had to pitch it up the Party/state hierarchy, and only once they has convinced the right ministry that they were worth implementing could their initiative be rolled out, from the centre. Innovations, for example in military or space technology or industrial processes, were largely state directed and assigned to bureaucratic ministries, and new ideas had to receive Gosplan approval (USSR State Committee on Planning).

Gosplan itself provides an excellent illustration of the process of Soviet innovation and entrepreneurship, detailed in this excellent paper on the creation and deployment of its first state computer, shifting the entire planning of the Soviet economy from a command driven to a data driven paradigm. You can see the sequence of ideation, political selling, leadership buy-in, and large scale implementation.

A more detailed glimpse into scientific intrapreneurialism is this fascinating deep dive into two of the pioneers of Russian computer science. You can see the process of getting to test and apply ideas for innovations. One of the innovators reads a Western text on cybernetics in a secret library in the Artilliery Military Academy in 1951 (cybernetics was considered a pseudo science by the state at this time). He then incorporates these ideas in his own sphere of influence, while publishing in the main ideological journal and giving lectures in Moscow, leading to promotions and the authority to implement his ideas on a bigger scale, with further publications and lectures and networking.

These two I think are excellent examples of the dynamics of innovation and intrapreneurship in the Soviet Union.

Where in the West that computer might have been built and promoted and sold, in the Soviet Union it was politically sold and promoted, but then tested at a scale most Western entrepreneurs would dream of for their very first iteration. The state was the sole Venture Capital funder, and evaluated the proposed ventures for political, then economic returns.