I want to ask several questions about the same problem domain but each question is independent from the others. Should I put all the questions in a single post or should I split them and send multiple posts?
Split them and ask multiple questions.
That way, it's easy to select a correct answer. If you ask several questions in one question you risk having answers that are both correct and wrong at the same time.
The second reason for splitting is that users, who will look for answers to similar questions later, would not likely have the same combination of multiple questions. Most likely, they would be looking for an answer to one of the questions and it would be more confusing and difficult for them to look for answers intermixed with answers to multiple questions.
Finally, it is in your best interests to split up the question. A single question containing many questions can be quite daunting and will deter people from providing answers - anyone answering will feel required to provide an answer to every part of your question in an attempt at giving the correct solution. By splitting your question you are much more likely to get positive responses.