1. It would be permitted to work on the check out of a supermarket that sells things Muslims consider impermissible (such as pork, alcohol, wine, or food items with haram ingredients). One’s earnings will be lawful (halal). This is the position of Imam Abu Hanifa (Allah have mercy on him).
One should, however, dislike such items in one’s heart: loving that which Allah loves and hating that which Allah hates is from true faith. (For the reasoning, see first appended answer, below.)
This ruling refers to doing so in a non-Muslim country or in non-Muslim areas in a Muslim country.
2. It is not permitted to sell such items in one’s own supermarket. If one does, it is sinful. The revenue generated from such sales is unlawful. It would be obligatory to:
(a) repent from one’s error itself (through remorse, seeking forgiveness, and determining not to repeat the sin), and
(b) giving away the unlawfully earned money to the poor. (See second answer below.)
3. It is not haram to touch unlawful items, especially when there is a need. Normally, meats are wrapped, and the alcohol in bottles.
As such, there will be no transfer of filth (najasa) onto one’s hands or body. Given this, there is no need to renew your ablutions (wudu) before praying. It may be best, though, to wash your hands, when it is reasonably easy to do so.
And Allah alone gives success.