Here’s a couple I thought of for your first request:
Uncertainty by Azeran Summary: "Do you ever stop and think about the fact that Loki tried to off himself?" http://archiveofourown.org/works/3513677
Freefall by galaxysoup – canon divergence Summary: Loki falls. Thor catches him. This does not solve as many problems as Thor thought it would. http://archiveofourown.org/works/346736
For a longer story, chapter 17 of this story deals with this issue: https://www.fanfiction.net/s/7876901/1/Truthfully
Also, chapter 29 of Lise’s “Remember This Cold”, “but I am hellbound” Summary: It wasn't a secret, exactly, but Loki hasn't discussed what exactly happened on the Bifrost with anyone. Neither, for that matter, has Thor http://archiveofourown.org/works/4826933
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Uncertainty by Azeran
Summary: "Do you ever stop and think about the fact that Loki tried to off himself?"
http://archiveofourown.org/works/3513677
Freefall by galaxysoup – canon divergence
Summary: Loki falls. Thor catches him. This does not solve as many problems as Thor thought it would.
http://archiveofourown.org/works/346736
For a longer story, chapter 17 of this story deals with this issue:
https://www.fanfiction.net/s/7876901/1/Truthfully
Also, chapter 29 of Lise’s “Remember This Cold”, “but I am hellbound”
Summary: It wasn't a secret, exactly, but Loki hasn't discussed what exactly happened on the Bifrost with anyone. Neither, for that matter, has Thor
http://archiveofourown.org/works/4826933