I've been monitoring PSRS for about 6 months and Fife PD has always been here, and always patched with TG40250 and TG40266. I occasionally monitor SS911 and have never heard Fife PD on that system, although I see that there is a "link" talk group listed in the RR database. If I had to guess I would say that it might be used for mutual aid for the agencies on SS911 to communicate directly with Fife PD.
The answer to your second question can be found on this page:
Seattle-King County Public Safety (KCERS) Trunking System, Seattle, Washington - Scanner Frequencies
I was just confirming no traffic on SS911. I have only heard them on PSRS. AD7TH stated he hears them all the time on SS911. Just wanted to make sure as I haven't monitored the Pierce county system for very long.
As for PSOPS S3, and PSOPS S4, they should be on one or both of the Pierce county systems. These talkgroups were on the old analog Tacoma system. They would patch one of those to whatever agency needed outside assistance. More than a decade ago Federal Way responded to assist Tacoma and were told to switch to PSOPS S3. Using the Trunker program I monitored the Tacoma system and found the Talkgroup. It was patched into one of Tacoma's tac talkgroups. This was when protests were blocking the Port of Tacoma military equipment shipments.
When Federal Way officers switched to PSOPS S3 their radios went on the Tacoma system. Likewise I would hear Tacoma officers come on the Federal Way talkgroup to meet up. This tells me they had neighboring county systems in their radios.
The King County system you linked to has PSOPS N1, PSOPS N2, PSOPS S1, and PSOPS S2. No PSOPS S3, or PSOPS S4. There are also PSOPS N3, and PSOPS N4 that I believe are on the Snohomish county system.
The next time I hear a reference to PSOPS S3, or PSOPS S4 I will monitor PSRS in search mode and hopefully catch it. It is a busy system so there will be a lot of traffic to sort thru.
To make things a little more confusing I also hear the PSOPS being referred to IO. I believe they are changing the names for the new upcoming King county P2 system.