- "With that card, I can travel with my Slovenian wife to the Canary Islands?"
Yes you can.
Visa required, except for Passengers with a residence permit
issued by Slovenia for a maximum stay of 90 days [in a 180-day period].
Doesn't matter if it was issued on the basis of a family relation.
- "Our flight is from Ljubljana with transit in London Heathrow Airport, where we have to collect and recheck our luggage. Do we need a transit visa?"
No, thanks to your Slovene residence card (for family members) you can stay in the UK together with your wife for as long as you want
Visa required, except for Passengers with a "Family Member"
Residence Card marked "Permanent Residence Card of a Family
Member of a Union Citizen or Residence Card of a Family
Member of a Union Citizen issued by Slovenia to family
members of an EEA national or national of
Switzerland if traveling with or traveling to join the EEA
national or national of Switzerland.
So just present your passport and residence card to Uk immigration