"They like us."
Translation:De tycker om oss.
They mean the same, although "tycker om" is slightly more formal. For example, I'd always use "gillar", but there is no right or wrong. Just keep in mind that "tycker om" could also translate to what you think of something.
Vad tycker du om honom? = What do you think of him? - As in "What is your opinion on him", not to be confused with the actual act of thinking.
I prefer mostly writing "tycka om", but in speech I use both. The reason you hear "gilla" more often is, because it is a slightly stronger form of liking and people usually talk about things they like or don't like. Tycka om is the lesser form of liking, so people are less likely to talk about things they only "tycker om" and more likely to talk about things they "gillar". An example of the difference could be a person who likes dogs, but likes cats even more. So when he talks about dogs he could say "Jag tycker om hundar" as if for cats "Jag gillar katter"
I'm not very knowledgable about Finland Swedish, though I wish I was. To be clear: this course only aims to teach Sweden Swedish. Hence, any information about Finland Swedish is very, very welcome in comments, but please mention when you write about it that you talk about Finland Swedish specifically, so learners know. Thanks! :)
That is unfortunately the system making an annoying error. The spelling dom is a colloquial but common and mostly acceptable spelling for both de and dem, since they're both pronounced like that. It should not be used outside of informal text, though, and is not actively taught. So the system should not show it to learners except for marking them correct when using it.