That's interesting. It is used in American English when someone uses the bed as a seat while eating, say if someone comes home and makes something to eat and sits on the bed to watch a movie, "Dude, don't eat on the bed." Eating "in bed" is usually used moreso if someone never left the bed or is staying IN the bed (vs ON the bed with the bed made) and has "breakfast in bed" or is sick.
Because "кровать" is feminine and ending in -ь. So -ь changes to -и.
No one would say that, but some would refer to sitting on the bed above the covers and eating as "eating on the bed." The course creators may be trying to distinguish between в and на here but perhaps when they review your report they will accept "my sister eats in bed."
есть in the infinitive form means both 'to eat' and 'to be' - the 2 words are identical in this form. However, if it's in a conjugated form such as ест, it can only mean 'eat'. (он или она ест = he or she eats). I guess моя сестра есть на кровати would mean she is on the bed.
1) You wouldn't write моя сестра есть на кровать, that would sort of mean "My sister there exists on the bed" but is nonsensical.
2) If it were a legitimate sentence you could still tell them apart though because ь slightly changes the sound of the consonant that comes before it. In the case of т, т without ь sounds similar to t in tar, while ть sounds similar to t in tee.