Rollovers happen at midnight est game time. The game time reads at 5:46pm as I am typing this out. Once that hit 12:00am? Rollover happens, but it can take time for it to fully finish. I tend to wait up till 45 minutes past, sometimes it finishes earlier, for it to fully get done.

Yes, your horses will continue to age. They technically age one year, in one real life week. So, if you are gone a week? They'd only age up one year. 

If you want to bypass this, you can freeze your horses via tabs underneath Estate: Manage. You will see an Overview, Tags, Tabs. Tabs is where you want to be at to individually freeze specific tabs or you can go to Account: Settings. Remain on the Profile tab and scroll all way to the bottom and there will be a big red button that will Freeze your whole entire account.

So what is the order of the tab for? I couldn't freeze my tab without it

For some reason you need to label each tab a number, so type in 1 to save and go through with 2, 3, 4 until you are done or you could just do 1 for each tab. It helps with organizing. Like if you peep at my profile to see the tabs? All are numbered and the ones at higher numbers, are towards the botttom and lower numbers are closer to the top.