For the query
db.products.find( { key: "a" } ), which one is more performant?
Given that field key is indexed in both cases, the complexity index search itself would be very similar. As the value of a would be hashed, and stored in the index tree.
If we’re looking for the overal performance cost, the hashed version would incur an extra (negligible) cost of hashing the value of a before matching the value in the index tree. See also mongo/db/index/hash_access_method.h
Also, hashed index would not be able to utilise index prefix compression (WiredTiger). Index prefix compression is especially effective for some data sets, like those with low cardinality (eg, country), or those with repeating values, like phone numbers, social security codes, and geo-coordinates. It is especially effective for compound indexes, where the first field is repeated with all the unique values of second field.
Any reason not to use hash in a non-ordered field?
Generally there is no reason to hash a non-range value. To choose a shard key, consider the cardinality, frequency, and rate of change of the value.
Hashed index is commonly used for a specific case of sharding. When a shard key value is a monotonically increasing/decreasing value, the distribution of data would likely to go into one shard only. This is where a hashed shard key would be able to improve the distribution of writes. It’s a minor trade-off to greatly improve your sharding cluster. See also Hashed vs Ranged Sharding.
is it worth to insert a random hash or value with the document, and use that for sharding instead of a hash generated on the _id ?
Whether it’s worth it, depends on the use case. A custom hash value would mean that any query for the hash value would have to go through a custom hashing code i.e. application.
The advantage for utilising the built-in hash function is that MongoDB automatically computes the hashes when resolving queries using hashed indexes. Therefore, applications do not need to compute hashes.