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Yes, you can collect and store the bulbils over winter and plant them outside in the spring. Place them in labeled ziplock bags. Then put them into the fridge for about a month or so to give them a cooling period. Take them out of the fridge and transfer them in growing trays or pots, then place these under Florescent lighting to grow them over winter. By spring the bulbils should be adequately sized for planting in the garden.

Yes, if you live in a suitable area where the threat of freezing temps is not an issue, then it is fine to divide and transplant both iris and lily plants. Just make sure they are transplanted in similar conditions, water deeply and trim back the foliage to lessen transplant shock.

Early fall is the best time to divide and transplant lily plants to thin them out.

Grow them outdoors in (Zones 7-10) shady areas in fertile, well-draining but somewhat moist soil. Indoors, or in pots, the plants require a peat moss and perlite mixture (or sphagnum moss) with bright, indirect light. They must also be kept moist, but not wet. They also need hight humidity, so daily misting may be necessary.

You did not mention which kind of bulbs you planted, but if they were like tulips, daffodils and hyacinth, then these bulbs cannot be planted in the ground year round where you live, as they need to be pre-chilled to be able to flower. This article will explain more about that: https://www.gardeningknowhow.com/ornamental/bulbs/bgen/how-to-plant-bulbs-in-the-south.htm It also sounds like they may not have been planted deep enough if they are heaving out of the ground. As for the lilies, you can just divide them and they will perform better when not crowded. This article will help with that: https://www.gardeningknowhow.com/garden-how-to/propagation/propgen/dividing-plants.htm