I am not a mechanic, so take it accordingly.
Before answering your points, make sure you study the relevant chapters of the Technical Manual (you are a Full member, you can use all resources):
https://www.sl113.org/wiki/Engine/ThermoTimeSwitchhttps://www.sl113.org/wiki/Restricted/Engine-starting-aid-tour#startingaidshttps://www.sl113.org/wiki/Electrical/WiringDiagramwhere the colour digram shows all the connections you need to see.
a.d.1. Yes, the Cold Start Aid system gets power during cranking only. That is why it is a Thermo TIME switch, it provides ground for certain time, not just for an instant, depending on temperature. It actually works not only for cold start. It works for all starts, just works shorter if engine is warmer. Thermo Time switches have descriptions about how much time they provide connection depending on temperature. Some owners, experiencing difficulties with proper functioning of the Thermo Time switch, install manual switch to have control over time the connection is provided.
a.d.2. The Thermo Time switch function is to enrich mixture o help start for certain period of time that depends on how warm is the engine. It is activated with crank signal and then keeps connection for some time depending on temperature.
G is not ground! This is critical. G is + to heat it up, then ground is provided from TTS housing to W to activate relay. Yes, ground is provided to W for several seconds depending on temperature. If your TTS does not do it, it is shot.
a.d. 3. I think you have starting aid system where this second relay operates fuel enrichment solenoid on your injection pump. In some cars it is not connected, e.g. I do not have that solenoid on my injection pump. It works in a very similar way as relay for cold start valve, but it works only when you crank the engine, not longer.
a.d. 4. I see on the diagram that wires to the enrichment solenoid are balck/pink for signal (from relay pin 87) and brown for ground.
Note on the wiring diagram:
20 - relay for Cold Start Valve
21 - relay for enrichment solenoid on injection pump
31 - Cold Start Valve
32 - Thermo Time Switch
33 - Enrichment Solenoid
On your points:
a.d. 2. Relay terminals
30: correct
85:
NO! Ground from TTS is from W terminal! Be careful there.86: Power from starter - yes;
but TTS connection - NO! goes to G terminal to warm up TTS, not to W. Do not swap these - you may cause fire, the circuit 50 is not fused!87: correct. To Cold Start Valve on one relay, fuel enrichment solenoid on the other relay.
a.d. 5 Connecting relay.
It does not matter if you trade sockets in the plug to change 85 to 86.
But it does matter that you do not connect + from fuse or starter to W.