I am a fairly regular diver in the Red Sea. Mainly liveaboards, but some shore diving too. Yes the Red Sea is safe to dive.
I would say that with 16 dives and travelling solo a liveaboard could be a big step up on your current diving. I would suggest a bit more shore diving before a week on a boat. Heading south from Port Ghalib probably gives the easiest liveaboard diving in the Red Sea, 'Wrecks and Reefs' is generally fairly relaxed diving too. I took my daughter on the southern route when she had about 25 dives and she coped fine - although that was a family trip aimed at novice divers
Lots of people do suffer with seasickness. With experience you can minimise it a bit (try to do very little when the boat is rocking) and over the counter seasickness tablets help a lot. After a few days on the boat everybody does get their sealegs and seasickness is less of an issue. Even people with really bad motion sickness do adapt in a day or two. The problem then is getting landsick when back on dry land...
If you want to go down the liveaboard route I would suggest dropping Emperor Divers an email and asking them a few questions to find a trip that might be suitable. They can probably advise which of their boats will have a higher proportion of newer diver on.
If you want to visit the Red Sea on a budget and do some good shore diving I would highly recommend Steve and Emma at Dive UK Hurghada. Only a small setup but offer good quality personalised diving. They have some deals on at the moment for dive and accommodation packages, not sure if they have them on their website or just Facebook.
I am not far from London, but now my only UK diving is in the swimming pool. Heading back to the Red Sea in July.