Hallo James,
A combi would probably supply your hot water and heating needs OK with just the three of you, but incorporating solar and the stove would be more difficult and more expensive. The regular boiler is vented to the header tank, or in reality the pipework very close to it is, the exact position of the feed and vent pipe is critical to the wellbeing of the whole system. The only gas boiler that supplies hot water on demand without a cylinder is a combi type, nearly all of which are designed to run a sealed and pressurised system. Personally I don't like vented systems, they are prone to problems that may go unnoticed or ignored which cause all sorts of damage. A sealed system shows up any problems of this sort with a pressure drop, which forces the issue.
Finding a plumber who can take on your system may be quite difficult, it may be worth specifying what you need from your system and then getting plumbers to cost and design a system for your perusal, then you could post up those suggestions here. There are many ways to set up a system to combine solar/woodburner/gas and all will have their merits and drawbacks, but without looking at your specific layout and demands it is quite difficult to be specific.
In order of complexity and cost, ROUGHLY your system would go:-
1 Combi and dry woodburner no solar
2 Combi and dry woodburner solar with preheat to combi
3 Regular/system boiler and dry woodburner solar and hot water cylinder
4 Regular/system boiler woodburner with boiler and solar with hot water with dunsley or system link
5 Regular/system boiler woodburner with boiler and solar with hot water cylinder and thermal store.
Or something similar, you could start from 4-5 grand for option 1 up to 15 grand or more for option 5

Lots of homework to do, but it is worth it to get your system right.
Desp