What are the design principles that promote testable code? (designing testable code vs driving design through tests)

Question

Most of the projects that I work on consider development and unit testing in isolation which makes writing unit tests at a later instance a nightmare. My objective is to keep testing in mind during the high level and low level design phases itself.

I want to know if there are any well defined design principles that promote testable code. One such principle that I have come to understand recently is Dependency Inversion through Dependency injection and Inversion of Control.

I have read that there is something known as SOLID. I want to understand if following the SOLID principles indirectly results in code that is easily testable? If not, are there any well-defined design principles that promote testable code?

I am aware that there is something known as Test Driven Development. Although, I am more interested in designing code with testing in mind during the design phase itself rather than driving design through tests. I hope this makes sense.

One more question related to this topic is whether it's alright to re-factor an existing product/project and make changes to code and design for the purpose of being able to write a unit test case for each module?