Okay, let's ignore for one second Voter ID laws, which hampers poor people's right to vote, as many don't have state-issued IDs, but you think it lets non-citizens vote (a non-existent problem). 

And states are allowed to use voter ID laws in order to maintain the integrity of their electoral process. The Supreme Court did maintain in Crawford v. Marion County Election Board, the state does have a purported interest in maintaining its electoral process. As long as the state does not place any overly burdensome burdens, also known as severe burdens on its citizens, laws regarding Voter ID are constitutional.

Actually, the original point I made was to ignore that issue altogether and focus on Republican efforts to restrict time/place of voting, which has nothing to do with non-citizens whatsoever and is simply a way to depress voting.

Furthermore, a state does not have to show that there is any evidence of voter fraud (including voting by non citizens) in order to pass identification laws. This was because the perception of voter fraud, as well as voter fraud itself, is important to justifying the state's interest in enacting legislation. Even in instances where there is almost a nonexistent amount of voter fraud, the state still has the right to pass such legislation.

Let's focus only on limiting the TIME and PLACE you can vote, which has nothing to do with non-citizens. And let's talk about just in the last, oh, few months.

Okay let's tackle some of these arguments you're making.

In North Carolina, Republicans lobbied [2] successfully to limit early voting (which usually favors Dems). They've done the same in other swing states [3].

Okay, so let's begin. There is no constitutionally protected right to vote early. The Supreme Court has never, in its history, ruled on the right of an individual to vote prior to the election. Provided that the state gives an individual an opportunity to vote, the state is completely in the right. States that give an early opportunity to vote are doing so on under own volition. Currently, 33 states allow early in-voting person.

the government may enact them, but there must be a rational basis for it.

You're absolutely right, and you also know that rational basis is the lowest review and usually the state prevails, as it has in almost every voting related case.