I'm a first year PhD student with mathematical finance background, and am not quite familiar with all the assumptions/setup for macroeconomics. During study I found the math relatively easy but I'm having hard time to grasp the often critical relation between variables. For example capital market clearing implies that the total capital that firms have equal to the total assets that households have. First of all, I didn't even know there's a market clearing until I ask my TA, secondly, there are too many such assumptions I don't know. My professor would probably think it's too dumb to tell students about these because most of my classmates have an Econ background, but I don't and it's really killing me. Is there any good reference I can read, open courses I can watch that could help me with this? Given limited amount of time, I hope it won't take as much time as a full course. Thanks in advance.
Surprisingly, I think the best way would be to read a Microeconomics textbook. You described difficulties with concepts like market clearing conditions. Macroeconomic models are usually based on general equilibrium models which students of Macroeconomics learn in their Microeconomics education. Consider reading the relevant parts of an intermediate Microeconomics textbook or Mas-Colell, Whinston + Green's classic Micro textbook for PhD students.
With this background knowledge, you can start reading Macroeconomics textbooks, e.g., Ljunquist/Sargent which teach you about the Macro-specific variations and applications.
You should also think about the quality of your program. A descent Economics PhD program will most certainly offer/require courses in Micro, Macro, and Econometrics.
Sorry if this sounds harsh, but:
You need to start asking more. You're not stupid for asking questions - you're stupid for not asking questions. Be upfront about your different background and ask away.
Of course you should try to research things by yourself, but a lot of assumptions are next to impossible to know or find out in a reasonable amount of time. I've got a 5 years of university economics - primarily in macro and I still struggle with assumptions. So, if you can't find it - ask! And ask your professor about some relevant books / articles.
Ahh, that's what I felt and was in the same situation for a year or so. What to do? I have accounting and finance degrees, BA and MSc, but continued education on a phd in econ. I started watching micro and macro lectures on Youtube, i.e. opencourseware channels, now, four year on, I record my own econ videos for my econ students. You just need to ask your fellows, many of my fellows, though they had econ degrees, were not able to answer simple econ questions. But I had to do my homework of watching videos and reading books. The steepest learning curve plotted when I started teaching as TA undergrad econ modules. You should get a part time TA job if you can, helps a lot.