If the shuttle hits the player then lands, what is the line judge supposed to call?
If the player then hits the shuttle out on their own side (does not reach or go past the net), or hits it directly down, what should the line judge call?
The responsibilities of the umpire and line judge are laid out in §17 of the laws:
§17.4 A line judge shall indicate whether a shuttle landed ‘in’ or ‘out’ on the line(s) assigned.
§17.6.1 An umpire shall (..) call a ‘fault’ or a ‘let’ should either occur;
The ITTO clarify further:
§8.7 [Line judge's] calls or signals shall always be made without any anticipation of the Umpire's decision regarding faults (e.g. the shuttle touching the player, his clothing, or racket before landing on the court, however obvious).
If the shuttle touches the player (or any other fault is committed), the line judge should observe the umpire (and during service, the service judge).
If the umpire or service judge calls fault or let, then the shuttle was out of play when it touched the ground, and no line judge signal/call is given.
If the umpire/service judge does not call fault/let immediately, then the line judge should signal and call In, Out, or Unseen, as usual.
There is one important exception: Per §220.127.116.11 of the ITTO, the umpire may omit calling obvious faults, specifically when the shuttle is caught by the net during the service, or fails to pass over the net. So if the fault is on the player's side and really obvious (e.g. shuttle is hit into the net, or deflected in a 90° angle during a defense attempt, or as in your question, a player hits the shuttle straight down), then the line judge should not signal anything.
That's what's written down in the laws and ITTO. In practice, the situation is a bit more complicated: At many international tournaments, the referees instruct the umpires to not specifically call out Fault for obvious faults, above and beyond what §18.104.22.168 ITTO allows, as this makes for a better presentation.
This should be mentioned in the line judge's briefing, and then the line judge doesn't signal anything either if the fault is obvious. But if in doubt, the line judge should always proceed with the signal, and let the umpire sort out the situation.
The rules and ITTO make it abundantly clear that only the service judge and referee can intervene if the umpire makes a mistake such as missing a fault, although all technical officials work together as a team (compare §6.9 and §7.8 of the ITTO).
The most a line judge can do if a fault has been missed is throw a slightly curious look towards the umpire, who will hopefully notice their mistake.