Perfectly understandable to feel anxious about your new job. I did just before I started mine, for the same reasons. Also, I changed companies a while back and was really agonising over the decision - will I like the work etc. Similar feelings!
Some depots have progressive links (rotas), where the junior drivers start in the lowest link with the worst work. Some have different links but are non-progressive - a bit of good and bad in each. Some depots just have one big link with everything in it.
Nights. I used to work at a depot with nights about every six weeks. Most of the work on nights was on the maintenance depot moving trains around. There was a bit of 'dead time' in the middle which was good for a bit of rest.
That depot has depot drivers now so nights isn't so often. The most frequent nights I know of on passenger work is at Bristol on the High Speed links, and Eurostar believe it or not, again because of moving trains around the depots.
Freight is a lot of unsocial hours - container trains are probably the least worst.
Where I work now has nights about every ten weeks. The most nights in a row would be five. It's a four day week but Sundays are in addition, so the Monday to Thursday with Sunday at the beginning would be longest run. Otherwise it's two, three, or four in a row unless you work your rest days. Some people really like nights. At every depot I've worked at people have swapped to do them - particularly if they live a couple of stops down the line - they can come to work by train on the last train, and catch one of the first back. Also, if they've swapped to nights a couple of weeks in a row then they can catch an extra Sunday or Rest Day and earn extra. This is quite common with people who like nights.
London Underground tend to have seven night shifts in a row followed by a week with more rest days in. There's fewer weeks of nights in the roster that way, you get them over and done with in one long week.
Whilst working shifts can be very tiring sometimes, there are times when it's brilliant. Seeing your kids school plays and sports days can be possible if your start/finish time is right. Or you might have that day off in the week.
It's a good job. It isn't for everyone of course, but if you've been trying to get it for this long, I think your anxiety will soon subside. Enjoy your training!