Are shingles contagious at all times?

Not contagious, but. Shingles is not contagious, but the virus that causes shingles (and chickenpox) can be transmitted in the fluid from the shingles lesions (not by coughing or sneezing). If these viruses are transmitted to someone who has never had chickenpox or has not received the varicella vaccine, then that person could develop chickenpox. The risk is much lower than the risk of transmitting chickenpox.
Yes, by contact. Chicken pox and shingles are due to the same virus (the zoster virus). If a person has no good immunity to the zoster virus (never caught chicken pox, is immunosuppressed, never got the vaccine, or got the vaccine but did not make enough antibodies) then he can get a case of chicken pox from the virus in the open, moist skin lesions of the shingles patient.
Yes. But not by droplets the way chicken pox is tramsmitted. If you touch th fluid from one of the shingles then get it in a cut or eye, mouth. Or nose and you have never had chicken pox then you could catch actual chicken pox.
Yes, but.... Only if you haven't had chickenpox. Chicken pox is caused by the zaricella virus. It is the same virus that causes shingles. The virus stays in your system forever, but may or may not reappear as shingles as you get older. If you haven't had chickenpox, you can "catch" the virus now if you come into contact with someone with oozing chickenpox or shingles.