Get names of all keys in the collection

You could do this with MapReduce:

mr = db.runCommand({
  "mapreduce" : "my_collection",
  "map" : function() {
    for (var key in this) { emit(key, null); }
  },
  "reduce" : function(key, stuff) { return null; }, 
  "out": "my_collection" + "_keys"
})

Then run distinct on the resulting collection so as to find all the keys:

db[mr.result].distinct("_id")
["foo", "bar", "baz", "_id", ...]

With Kristina's answer as inspiration, I created an open source tool called Variety which does exactly this: https://github.com/variety/variety

You can use aggregation with the new $objectToArray aggregation operator in version 3.4.4 to convert all top key-value pairs into document arrays, followed by $unwind and $group with $addToSet to get distinct keys across the entire collection. (Use $$ROOT for referencing the top level document.)

db.things.aggregate([
  {"$project":{"arrayofkeyvalue":{"$objectToArray":"$$ROOT"}}},
  {"$unwind":"$arrayofkeyvalue"},
  {"$group":{"_id":null,"allkeys":{"$addToSet":"$arrayofkeyvalue.k"}}}
])

You can use the following query for getting keys in a single document.

db.things.aggregate([
  {"$match":{_id: "<<ID>>"}}, /* Replace with the document's ID */
  {"$project":{"arrayofkeyvalue":{"$objectToArray":"$$ROOT"}}},
  {"$project":{"keys":"$arrayofkeyvalue.k"}}
])

A cleaned up and reusable solution using pymongo:

from pymongo import MongoClient
from bson import Code

def get_keys(db, collection):
    client = MongoClient()
    db = client[db]
    map = Code("function() { for (var key in this) { emit(key, null); } }")
    reduce = Code("function(key, stuff) { return null; }")
    result = db[collection].map_reduce(map, reduce, "myresults")
    return result.distinct('_id')

Usage:

get_keys('dbname', 'collection')
>> ['key1', 'key2', ... ]