[Playground] Share any code feature frontend

[alexeyinkin]

Resolves #21978.

In this PR:

• Added 'share any code' feature.
• Refactored dropdown offset calculation to accommodate the new button.
• Refactored color theme handling because the new button did not fit the old theme handling.
• Added ExampleModel.sdk because there are now cases when we can't take it anywhere else, and it also feels natural.
• Fixed tests, they broke after adding sdk.

Many thanks to @miamihotline who did 70% of the job but was interrupted.

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[alexeyinkin]

R: @damondouglas

[github-actions[bot]]

Stopping reviewer notifications for this pull request: review requested by someone other than the bot, ceding control

[pabloem]

lgtm!