In the roman republic can everyone participate in government to some degree

Yes because all were allowed to vote, which is crucial in government

False.

Women and slaves could not.

Males (although I seem to recall there is something about the poorest of the poor being excluded from certain things in the Republic such as the army, perhaps politics) were divided into groups based on wealth and land: a pyramid scheme. These groups came to collective decisions and cast their vote. Although the largest groups contained the poor, and the smallest groups contained the wealthy ... so the rich, while the minority, could still cast more votes due to how the system was structured.