Swedish has masculine and neuter, but not feminine, interestingly enough.
Neuter has stayed as is.
There are, however, remnants of the masculine/feminine distinction left: e.g. klockan is ofter referred to as feminine (hon), and the "-e" in "storebror" (as opposed to "-a" which is used for utrum today and feminine in the past).
The -e/-a ending is still in common use not only in constructions like "storebror" but in general in adjectives pointing to a female or male person ("den fule mannen", "den fula kvinnan") although this usage is falling out of favour a bit ("den fula mannen" would not generally sound wrong to most modern speakers)
Another interesting remnant of old Norse that still lingers in modern Swedish is the accusative case ending -s in expressions like "till fjälls", "till skogs", "till sjöss", "till sängs" etc.
No, this is not accurate, neither historically nor according to modern usage. Animate objects tend towards utrum but plenty of inanimate objects are utrum too and there are no hard rules or mnemonics you can use to divine which is which.
It's "ett tåg" ("a train") but "en bil" ("a car"), "ett äpple" ("an apple") but "en banan" ("a banana"), "ett barn" (a child) but "en vuxen" (an adult). This is entirely arbitrary and non-obvious to non-native speakers and if you're learning Swedish, you just straight up have to learn which is which for every word.
For example, "a friendship" is "et venskab" in Danish, but "en vänskap" in Swedish.