Are there any "slow" sidechains? What are the tradeoffs? Are there any examples of sidechains which target a block time much longer than bitcoin? Could a sidechain be made "slow enough" such that the size of a proof (SPV proof?) for it could actually fit into and be verified by a mainchain bitcoin transaction with the currently-in-effect mainchain protocol rules?

I'm interested in learning more about how a sidechain with a targeted time between blocks of, say, 1 week, might work and/or if such a chain, if it existed, could be useful.

I'm specifically interested in proof-of-work sidechains that do not rely on any sort of federation (as such, Liquid and RSK would not meet this criteria at the moment, but I'm unaware of any other current examples of bitcoin sidechains), and I'm most interested in "slow sidechains," if any exist.
Are there any &quot;slow&quot; sidechains?

Softchains fit this definition. Softchains are a type of sidechain that are verified by all mainchain users in a slow but extremely resource-efficient way, by only validating the blocks for which forks occur. It's slow, because you can't assume these forks will occur right away. People who want to actively use the softchain perform full validation, which means consensus will be fast for them.

What are the tradeoffs?

These softchains are not fully decoupled from the mainchain. Creating a new softchain requires a soft fork, and softchain consensus could negatively affect mainchain consensus under some scenarios (software bugs, one-year reorgs). Furthermore, mainchain validation cost goes up by roughly 100MB per softchain per year (which could be compensated for with a mainchain block size decrease).
The above is just a high-level overview.
For a relatively easy-to-follow explanation, I recommend listening to this podcast:
https://youtu.be/4c7cwTts3fo?t=1m45s
And the full details are in this write-up:
https://gist.github.com/RubenSomsen/7ecf7f13dc2496aa7eed8815a02f13d1