Optimal triple town algorithm

I've been trying to find an optimal algorithm for a problem inspired by the game 'triple town'. The game goes as such:

You place objects in a grid, and each time you make a set of three, they condense into one object of higher level at the position of the last object placed.

Furthermore, if you place three of these b objects together they again compress to form an even higher level object.

Note: in these diagrams the level of an object is expressed as a_i, b_i, and c_i and the subscript denotes the objects number in the set of three.

To simplify things, I am only considering when every object you have to place is of the lowest level.

Now my questions are:

1: Is there an algorithm to determine the least amount of grid area needed to make an object of level x, given x?

For example, for level a you need 1x1, for level b you need 1x3, for level c you need 1x5.

2: Given the dimensions of a grid, can we find the highest level and number of objects achievable?

For example, for a 2x2 you can get 2 level 'a's and 2 level 'b's

3: Is there an algorithm to find the optimal order and position of objects to get the highest possible level, given a fixed grid?

For example, for a 2x2 you can get (1,1),(1,2),(2,2)

4: Given a position of an intended level x object, what set of moves minimize the amount of space needed to make this object?

5:What are the optimal complexities of these algorithms?

Update:

One thing that I think is prominent in the finding of solutions is that the getting an item of level x can't be done in any arbitrary p