I don’t like Lee Jong Suk either but for the shallowest of reasons - his nose really annoys me 
Funnily enough I didn’t think much of Secret Garden or Korean Odyssey after their first episodes. In both cases there were months between me watching their first episodes and going back to give them another try. They are both the only two programmes I’ve done that with and both are now my absolute favourites. For me the first episode of Secret Garden was a bit dark, but then it turned wonderful! Hyun Bin is hilarious and the Oska character, as i’ve Said before, is one of my all time fave 2ML. Personally once it gets to the body swapping bits it’s laugh out loud funny.
My username - Heechul Oppa - is because Heechul is my oppa! Kim Heechul is in Super Junior (Siwon’s group) and he is also one of the hosts of Men on a Mission. He’s hilarious and very unpredictable by Korean celebrity standards. If you watch the Mr Simple MV he’s the third guy to sing solo - the one in shades with a ponytail.
My dh loves Kdrama and kpop as much as I do. He actually got me into it! Basically, there was a Japanese actress he likes (Haruka Ayase) and he’d heard she’d been in a funny Jdrama. We have a multi-region DVD player so he bought a copy of the drama (Hotaru no Hikari) and we adored it! So we bought the second series and loved it to. Then I found a streaming app that let us watch jdramas so we got totally hooked.
Then the app disappeared!!
We were genuinely gutted, then dh found Viki, which has a few JDramas on it (I really recommend Last Cinderella). But they don’t have many Jdramas. Dh was off work around that time so he watched the first few episodes of Secret Garden and Love Rain and really enjoyed them, so persuaded me to try them. As I mentioned I thought Secret Garden was too dark (JDramas tend to be even fluffier than kdramas and the themes tend to be more ‘slightly older lady left on the shelf, gives up hope of love, bags a younger boyfriend). Plus I really missed hearing Japanese and seeing the Japanese culture (which is very different to Korean culture. Korean attitudes are more similar to western attitudes I found, but that’s by the by). Anyway, I ended up being hooked by Love Rain, went onto Gong Shim and a few others, went back to Secret Garden and was absolutely irretrievably hooked!
Re Siwon’s dog... er, I suspect his ‘rehoming’ might have been a euphemism, yes.
Ah Bea, it’s really sweet you think you won’t go down the kpop rabbit hole! I was once like you too. What you don’t realise is that you already are! Most dramas have kpop soundtracks, that’s how it gets you. And basically all kdramas have at least one kpop star as a lead / 2nd lead. Gong Shim herself, Siwon, I think the FL in Secret Garden, V (from BTS) in Hwarang, The annoying try-hard guy in Secretary Kim - list goes on.
Embrace it!