Yes the mailman or mail lady can legally refuse to deliver your mail to you.
If the mailman or mail lady tries to deliver your mail and there's something that can harm the USPS worker such as a vicious dog or other hazard then they can refuse to deliver your mail.
The postal workers do not have to deliver your mail to your address and so you would have to fix the issue before they would deliver your mail or if possible get a post office box.
USPS does deliver mail on Saturday.
Saturday is a business day for USPS so since they deliver mail to the post office the USPS will also deliver mail to your curbside mailbox or house.
The actual time for Saturday USPS mail delivery varies but for my house I get my curbside mail through USPS on Saturday by 10:00 AM and on Monday through Friday my mail usually gets to me by 8:00 AM to 9:00 AM.
The USPS sorting facilities are open everyday of the week 24 hours per day sorting mail and sending it on it's way.
But the local post offices are closed on Sunday and Federal Holidays but the mail is still being sorted and transported.
You should get your mail on Saturday if you're expecting mail.
But the mail must get on the postal deliveries truck before they leave out which is usually around 7:00 AM or before.