Just like physical stores, most of the answers to your questions will vary depending on which online vendor you're using. Some are better than others.
"Safe" is hard to answer. Are you in a remote area? How do the different shipping services do in terms of reaching you on time? With fish shipments, there's really no room for delays. If your UPS Next Day Air delivery turns into 48 hours because you're in the sticks and weather was iffy, it's not safe at all. But if shipping is generally good/on time, you'll probably be alright. Just be cognizant of the time of year (avoid ordering during the holiday season as delays are common) and the weather (try to avoid ordering if it'll be shipped during extreme heat/cold or weather that could cause delays).
Fish health depends entirely on the vendor. Personally, I have only purchased through vendors that have a quarantine process.
Dr. Reef is my personal favorite. I also like
TSM Aquatics and I've had luck with
Diver's Den (though I rate it 3rd of this bunch and avoid the standard Live Aquaria section). On top of treating fish for normal ailments like ick, flukes, velvet, etc. (you should still QT anyway), buying pre-quarantined fish ensures that the fish is doing OK an an aquarium environment (not hiding a disease, eating, etc.). It's not perfect, but it's a safer bet than most other options. Some online stores (like Divers Den) have a "What you see is what you get" option if you want to actually see the specific fish you're getting.
