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On average, most plastic containers retain a lot of moisture, which can lead to rotting of the sweet potato tubers. That being said, as long as you can ensure that adequate drainage is provided, growing these plants in a 5-gallon bucket should be fine.

These articles should help: https://www.gardeningknowhow.com/edible/vegetables/sweet-potato/harvesting-sweet-potatoes.htm, https://www.gardeningknowhow.com/edible/vegetables/sweet-potato/grow-sweet-potatoes.htm

Slips are rooted sprouts from mature sweet potato tubers. Yes, these can be grown in grow bags. Be sure to mix in a shovelful of compost with the soil in your growbag. After harvesting, they need to be used within a few months, as sweet potatoes do not store very long. Allow them to cure, or dry for a couple weeks prior to storing. For additional information, these articles should help: https://www.gardeningknowhow.com/edible/vegetables/sweet-potato/grow-sweet-potatoes.htm, https://www.gardeningknowhow.com/edible/vegetables/sweet-potato/harvesting-sweet-potatoes.htm

Sweet potatoes need lots of warmth in order for them to sprout. This article should help: https://www.gardeningknowhow.com/edible/vegetables/sweet-potato/grow-sweet-potatoes.htm

Sweet potatoes are not started in the same way regular potatoes are started. They need to be started from slips. Here is more information on that:
https://www.gardeningknowhow.com/edible/vegetables/sweet-potato/sweet-potato-plant-starts.htm

If you did start from slips, then your friend is right in that it was likely a nutrient imbalance. Nitrogen is great for leaf growth but too much can actually keep roots (and the tubers we like to eat) from growing. You need phosphorous to have healthy root and tuber growth. Next time, you can use the mushroom compost, but add in a healthy amount of something phosphorous rich, such as bone meal.

Bone meal and other phosphorous rich fertilizers are sold in most gardening stores. Follow the directions on the package and add it in right at the beginning.

Because these plants are actually not related, they sweet potaotes are not grown the same as regular potato plants. For container growing, you would plant your slips anywhere from 3-6 inches deep, burying them completely with exception to any top growth that may be present.