does automod work when the bot is offline?

yes for discord's own rules. fray writes them into your server, and discord runs them without fray.

can automod time people out?

yes, from one minute to one day. discord's spam and word filter rules can only block.

yes. exempt up to 20 roles and 25 channels. fray's own checks already skip anybody with manage server.

discord's own rules, kept in order: invites, links, spam, mention floods, swearing and nsfw words, blocked by discord before the message lands.
four checks on top: too many capitals, emoji, images, or forwarded messages, deleted just after they're sent.
nothing changes until you apply: setup walks through six steps and shows every rule it will make, keep or remove before it touches your server.
your exceptions: up to 20 roles and 25 channels automod leaves alone.

every option is on the docs page.

does automod work when the bot is offline?: yes for discord's own rules. fray writes them into your server, and discord runs them without fray.
can automod time people out?: yes, from one minute to one day. discord's spam and word filter rules can only block.
can staff be left alone?: yes. exempt up to 20 roles and 25 channels. fray's own checks already skip anybody with manage server.

fray covers moderation, tickets, giveaways, reposts, roblox and ai, so a server doesn't need a separate bot for each of them.