Haven't played CoC for some while, but I remember that shared replayes, just as chat messages, are visible in the log depending on the amount of messages (fewer messages => the replay is visible for a longer period). You can create a new clan and share there, but keep in mind that it can also disappear from there on a new update

For attacks and I think also for defense, you can only view attacks that have happened after the latest update.

I think they also contain only about 10 replays at a time each

Can I watch shared replays on Clash of Clans after season reset?

It depends.

Why: If a season reset hasn't had a new patch that change anything, then yes. Otherwise no. This is because "replays" are a replay of a the player's inputs, as opposed to what actually happened. Everything else that happens is basically a simulation of that same attack. As such, if mechanics or stats are changed between patches, it would invalidate a replay. This is why replays do not persist between patches.

You can always record a replay using a screen recorder. Most phones should have an in-built one with their respective OEM-specific Android flavor.

Do I have a time limit to watch shared replays?

Yes

See above; that and if it pops off your Battle History or Clan's Chat Log.