1) Why one is better? The battery is your limiting element for power.
2) Neither I nor anyone is trying to tell you "just do this." However, with your limited background, you either have to start with the basics and work up (Is there time?) or take some things as facts from those who have more experience than you.
3) I forget where you are located, if you ever indicated that. It makes a difference as to availability of materials (e.g., NiCd vs. NiMH). What grade level is this project for?
4) Look at this as an example in project management. Divide it into various essential parts and goals. For example, you have two goals: 1) Maximum one-time lift; and 2) Maximum sustained lift and release (20 s). And, there are easily 5 parts to each goal: 1) The battery; 2) The wire size; 3) The number of turns; 4) The core size and shape of coil; and 5) The switch.
Look at each one and decide which will be most difficult to source for, say, the sustained goal. Which will be hardest to change during the course of your development? For example, the battery and switch are easily changed and readily available. So, I wouldn't worry about either of them much right now. For the battery, you have essentially 3 options: alkaline, NiCd, and NiMH. Several of us have recommended the NiCd as having the highest current capability. You've argued against that. Just pick a battery chemistry and move forward. You can try others and compare the results.
If I were to look at the 5 elements, I would consider the magnet core as perhaps the most difficult item to source and change once incorporated into the design. So, I would start looking around for a suitable core. You know up front some of its general characteristics. It needs to be big, but not overly massive. Can you find a U-shaped core? Can you bend a 2.5cm dia. steel rod? Can you find a suitable cast iron core? Maybe you have an old microwave transformer, can disassemble the core, and convert it to a E-core? Or, maybe there is an electronics surplus store in your area and you can scrounge around to find something.
Those are all just suggestions. The message is to get organized, develop a "critical path" and get underway at solving the problem.
For the single lift, you can consider using capacitor discharge to increase your current by switching the cells to series. So, I would focus on the sustained carry first.
John