In the theme, Live search it is shown the sale price even if out of the scheduled dates. Should be displayed instead the regular price.
Another cool feature is to show the regular price crossed before the sale price.
1) Can you send me a link to your website and tell me which product I should search for, please?
I have tested it in my environment, and if the product is on "sale time" it shows the sale price, but when there is still a couple of days left till the sale, it shows the regular price.
2) Indeed, it sounds like a cool feature to show crossed regular price.
Please, share it with us also on the following topic, so we will not miss it.