After the idea arrives to the senate, who is it assigned to

If you mean bills ,Generally, bills are legislative proposals that, if enacted, carry the force of law, whereas resolutions do not. Though, this is not always true. ... A Bill starts either in the House of Representatives or in the Senate. Bills that start in the House are designated “H.R.”

The way this small state - large state compromise guards against tyranny is that small states and large states have one note: i am sorry but i was not able to give you the full answer. but i got this answer from