"We love them."
Both ich and je can mean "them" in the sense that it's used here. Ich means a male personal group (a group with at least one male person), and je means any other group.
"je kochamy" is wrong probably only for the same reason that "ich kochamy" is wrong (see above)
"kochamy je" should work though
If only you can do it another way, avoid putting a pronoun at the end of the sentence. It sounds really quite clumsy. For example here: "Kochamy ich" is a perfectly fine sentence (the alternative, putting 'ich' at the beginning would be really really bad), but "My kochamy ich" is clumsy, because you can easily say "My ich kochamy". Love is the most important information, not 'them'.
Yes, implied, but not explicitly put. As you use two words, the only two options are "Kochamy ich" or "Ich kochamy", and "Ich kochamy" is just too strange, you really shouldn't start with a personal pronoun in a different case than Nominative.
When you put this subject explicitly, the number of potential word orders grows. You can now not avoid putting 'ich' both at the beginning and at the end of the sentence, at the same time. Therefore you should do it :)
When translating "they", you have two variants: "oni" (at least one man among them) and "one" (no men among them).
"ich" is the Accusative form of "oni", while "je" is the Accusative form of "one". Although "Kochamy je" may be about some women, it also works perfectly for any other plural noun that doesn't describe male people.