I retired and asked to do a direct rollover from my former employer's 401(k) plan to my IRA. The check was mailed to my house but made payable to my IRA custodian. I just realized I didn't deposit that check right away. It's now past 60 days since I received it. Can I deposit that check now or is it too late?

You're in luck. You can deposit that check even though it's after 60 days from when you received it. The 60-day rule does not apply to a direct rollover. A direct rollover is when you don't have control of the money. In this case the check was made payable to your IRA custodian, not you. Therefore, you can deposit that direct rollover check after 60 days.

My Aunt died and I directly rolled over the pension money she left me to an inherited Roth IRA. Is this a tax-free rollover?

No. This is a taxable rollover because the money was rolled into an inherited Roth IRA. If you had directly rolled over the money to an inherited regular (Traditional) IRA, it would have been a tax-free rollover. The good news is that when you later take the money out of the Roth IRA, it will generally be tax-free.

I took out $20,000 from my IRA last month in one withdrawal. I'd like to complete the rollover of $20,000 with multiple deposits to my IRA during the 60-day rollover time frame. Will the IRS allow me to do that or must I roll over the entire $20,000 in one deposit?

It's your choice. You can roll over the $20,000 in one deposit or you can do multiple deposits totaling $20,000 on separate days, as long as the deposits are made within 60 days.