No, it does not allow concurrent writes (with some exceptions if you get into it [0]). You should generally use it only if write serialisation is acceptable. Reads and writes are concurrent except for the commit stage of writes, which SQLite tries to keep short but is workload- and storage-dependent.

Now this is more controversial take and you should always benchmark on your own traffic projections, but:

consider that if you don't have a ton of indexes, the raw throughput of SQLite is so good that on many access patterns you'd already have to shard a Postgres instance anyway to surpass where SQLite single-write limitation would become the bottleneck.

[0] https://www.sqlite.org/src/doc/begin-concurrent/doc/begin_co...

No it doesn't - it allows a single writer and concurrent READs at the same time.