Well out of those three options, I'd say a bank would be your best bet. A lot of the time some banks are almost exclusively meant for home equity loans. It would be worth checking out a bank, and I'm sure you'll find lower rates there. I'm hoping to buy a home for the first time sometime soon, so I'm right there with you. https://norwoodbank.com/mortgages/home-equity-loan-options/
It depends - some credit unions are better than others and require you meet certain membership criteria (US armed forces for example) or belong. Credit unions have a good rep because they can be more lenient whereas big lenders may reject a borrower. Mortgage lenders on the other hand shop other banks, rebate programs and other types of lender (including credit unions too) to find something that matches your situation that you may have no idea about or something not widely advertised. Mortgage brokers don't get paid until something closes so they have an incentive to work hard. In the end it's a draw because adding a mortgage broker into the equation adds another set of people in the mix if they end up finding you the same type of program you could find at your local or selected credit union that could just be a hassle. But on the other hand if the broker found something unique it's great to have more people working for you.