Cembrit are perfectly fine, i've used them many times.
You will need a bonding gutter if next door is still the original slate.
Replacement slate is easily available. You will have to weigh up the best option.
Best way is to back bed the tiles on with mortar.
Leave it overnight and then you can fill the rest of it up and the mortar will have something to key to. As it is at the min you are probably not pushing enough in.
I use mid range Cromar, the blue one. £44 a role and it is decent quality.
I'd strongly suggest against the cheapo 20 odd quid stuff as that is way too thin and damages easy.
Our way is to just hit the top corner of the slate once or twice and this usually snaps the nails. If the slate is no good it will break. Scaffold board in the gutter to catch them all. Stack, re-hole and grade.
Obviously this only works if the nails are knackered.
Please upload some pictures of said repairs if you are still alive in 3 months.
I won't even entertain this sort of work without a scaffold or tower at the minimum. Just not worth it to save the customer money.
Individual soakers and step flashing is for slate and small plain tiles really.
The job isn't as good as it should be but looks dry enough. It doesn't look like he owns a lead dresser!
A proper roofer is the best idea.