Learn Swedish: Jag ger kassen till kunden. - I am giving the shopping bag to the customer.

In Swedish, till is used for the physical recipient of an object. Since you are physically handing the bag to the customer, till is the correct choice. If you were carrying the bag on behalf of the customer (perhaps because it was too heavy for them), you would use åt instead.

What kind of bag is a kasse, and how does it differ from a väska or a påse?

A kasse is specifically a shopping bag or carrier bag with handles, like a standard grocery bag. A väska is a personal bag that you own, like a purse, backpack, or suitcase. A påse is usually a smaller bag, often without handles, like a little paper bag for candy or a plastic freezer bag.

Could I rewrite this sentence without using the preposition till?

Yes! Just like in English ("I give the customer the bag"), you can drop the preposition if you change the word order. You would put the indirect recipient (kunden) right before the direct object (kassen): Jag ger kunden kassen.

Why does kassen only add an -n to become definite, rather than -en?