No, any old spider egg will do. The spider eggs in the lower room are just if you haven't had the luck to encounter any yet. If you've got loads of them, just use them instead.
Other than that, the only way I can think past the grate is to jump into the crushing pit from the top level, drop the egg there, then walk out of the door into the small hallway, then flip the lever there and go back up to the top.
-Sazerac
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