It seems like your eaves are missing or broken in areas. There are 2 ways of dealing with this problem. You can install plastic eaves trays beneath your existing 1st course of slate, or your roof needs to be stripped out in a 'v' section to allow access for new slate eaves to be reinstated to your roof. The roof can then be re-slated with minimal lead tabbing visible.
You may need a lead saddle installing beneath your ridge. A lead saddle is an essential part of your leadwork when a home as a chimney that sits at the apex of the roof, and acts as a bridge between the front and rear slate soakers or tile flashings on your chimney.
Yes. A preformed fibre glass bonding gutter will need to be installed above the partition wall seperating both properties. You simply tile and slate both roofs to the center point of the bonding gutter and it will then act as a soakaway for any rainwater in between the 2 roofs. It's not visible, and is a very effective and neat solution to this type of problem.
Yes. By starting at the top of the valley and working down either side, enough slate/tile need to be removed (usually two) to expose the edges of the valley boards and the fixings holding the lead in situ. The lead can then easily be removed and replaced with new. The slate/tile can then be reinstated and the ridge bedded and pointed as before.
Obviously, each roof is different and taken on its own merits. Whether it's a slate, tile or fat roof there is usually something that can be done to repair and prolong the roofs lifespan. By replacing broken or damaged slate/tile or patching and sealing a flat roof you can achieve this. The problem will be the condition of the battens or the flat roofing boards, and whether or not they can still take the weight of the roofing products sitting on them. If they are rotten and in a dangerous state then you will be looking at a full re-roof unfortunately.
No, it will not last anywhere near as long as lead. Flashband should only be used as a temporary repair to alleviate water penetration while the correct programme of repair is being finalised, or you are waiting for a start date to renew your leadwork. Flashband will only last a couple of years weathering before it begins to crack and fail.