You need to review the instructions and the form very closely. Regarding the entity name, you must include the appropriate designation, whether it is Inc., Co., or something else.
If you are making an election of managment by membership, it sounds as if you are looking at a limited liability company rather than a corporation. You really need to consult with someone who can explain the fundamental differences between the entity types before you get into questions about member managed versus directors.
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Texas law does not require a corporate designator is not required in Texas but can be used. If you use it then it is your name.
Generally speaking - Members elect the board so you need to have two meetings - one for members and one for the board.
If you just have a board then it chooses the board elects the new board - I find this structure preferable.
No way of avoiding having a board either way.