How do I confirm this board fits my TV?

Check the model number on the TV's rear label against PN51E440A2FXZA and PN51E450A1FXZA, then compare the part number printed on your old board's barcode label with BN94-06039B. The board number BN41-01799B should also match.

Is this the same as BN94-04343J or BN96-24581A?

It is listed as a substitute for both. Treat that as a cross-reference, not a promise: confirm the connector layout and board number on your set before ordering.

Will a new main board fix my TV that won't turn on?

Only if the main board is the failed part. A dead set can also come from the power supply or the Y-sus/X-sus boards, so test those before replacing this board.

Does it come with cables or the back cover?

No. It is the main board only. Reuse your existing cables, screws and covers, and note the connector positions before you unplug the old board.

Do I need to update firmware after installing it?

Usually not. The board carries its own firmware, but a board pulled from another set may run a different version, so check the menu after installation.