(I started this reply while DollyDaydream was finishing up, so I hadn’t read their response yet. Expect possible repeats of what they’ve covered.)
In English, you might say “I like dogs.” In Japanese, it’s more along the lines of “dogs are likable” or “dogs are favorable”. (The sentence is an “A is B” type sentence.) Typically it’ll be clear who has a favorable view of dogs, but if not then は can be used to clarify.
The context is around the policeman, Inspector Kuroboshi, thanks to the は in the prior sentence. (「そこにでてきたのは警視庁の黒星けいぶだ。」) Because of this, we know that 犬がすき is in context of Kuroboshi. The subject of 犬がすき is 犬, but the one for whom 「犬がすき」 applies to is the person being talked about, Kuroboshi. (In the words, you are correct, for these reasons.)
The したことがある tells us (the narrator) did something before, and the ように tells use he tried to make the action ように is attached to occur. So (close to what you wrote), he (the narrator) once trained Inspector Kuroboshi’s Pomeranian to do a handstand. This sets up why Kuroboshi knows Spitz: Kuroboshi likes dogs, and Spitz previously trained Kuroboshi’s Pomeranian to handstand. Note that the dog doing the handstand isn’t doing so right this moment. The drawing is just picturing what happened in the past.
There are a few clues.
- できる leads into よう. This means you have a clause (I never remember the right term, but essentially a sentence embedded in a sentence) here. The embedded sentence is modifying よう.
- した leads into こと. The embedded sentence is modifying こと.
Here’s how I visualize these “embedded sentences” which modify the word that follows them:
The innermost sentence: “Trained that Pomeranian to be able to do a handstand.”
The next sentence going outward: I tried to [inner sentence above]."
And the outermost sentence: “Previously, (I) did [inner sentences above].”
And all together: “Previously, I had taught that Pomeranian to handstand.”
で is the て form of だ, so you are correct that it’s essentially “and” here.