It's unusual to have the wire on L2. But not a problem with the Hive receiver.
L to L
N to N
LX to 1
L2 to 2
L1 to 3
This is assuming that it is a single channel Hive that you have got.
In the UK we call it live (or line), Americans call it hot.
As to your first question, are both the lights on the same fuse/MCB?
If they are then you can just put the 2 red live cables together in the common terminal.
If they are on different fuses then you will need 2 x 1 gang switches.
Why not just move the hub to where the wiring centre is and wire it directly to it.
Where the old clock was just join the neutrals together in a connector, do the same with the lives in a separate connector, make the other 4 wires safe in individual connectors and fit a blank plate over them...
Ideally you need the single channel Hive, but if you put the grey wire into terminal 4 instead of 3 it will work.
Or depending on what clock you have you could do the job properly and replace that with your new dual channel Hive so it would control the hot water as well.
The controller may offer those alternatives but you are limited to how your system works.
The thermal store has to be heated up before the heating (or water) will work, this may take over an hour.
If it has been heated up and you are getting hot water but still not getting any heating then the...
With a thermal store for the heating to work the hot water has to be turned on too, you can't have heating on on it's own.
Depending on which controller you have it would usually be set to gravity mode so when you select heating it automatically turns hot water at the same time.
That looks like an automatic filler that the newer worsceter boilers can have fitted.
It should top up the pressure automatically when it drops below 0.7bar.
It will only do it so many times though. Then it will need resetting in the settings.