I’ve done some further testing on this. I’ve upgraded the Mac Pro, MacBook Pro and MacBook Air to El Capitan 10.11.4 (with associated Mail 9.3). The Mac Mini is still on 10.11.3 (and Mail 9.2).
On the Mac Pro and the Mac Mini, Spamsieve still has the original issue of opening, closing and then re-opening during booting up (and letting spam into the Inbox during its short closed-down time) - if it is in the Startup items. It is ‘cured’ by starting Spamsieve manually after booting up, and continues to work perfectly thereafter.
On the MacBook Pro and MacBook Air, Spamsieve works perfectly all the time, and is included in the startup items.
The main difference between these two sets of Macs is that the MacBook Pro and MacBook Air both have SSDs, and the other two have old-fashioned mechanical hard drives - and are slower at booting up.
I wonder if this gives you some hint about what is happening? It’s such a shame that this has only recently gone wrong, when it has been working excellently for years.
I think ‘TinkerTool’ used to have a feature of being to decide on the order of the Startup items, but apparently it doesn’t now make any difference about the order. Another open source application ‘DelayedLauncher’ might work, but I have no experience of this.