The mathy square strikes back!

Question

This is a follow-up question to Does this mathy square have any solutions? (And how many?).

Consider a 7x7 grid of math operators o and numbers A, B, ..., P:

A o B o C o D
o o o   o o o
E o F o G o H
o   o o o   o
I o J o K o L
o o o   o o o
M o N o O o P

This grid encodes 10 math equations. There are 4 horizontal equations, 4 vertical, and 2 diagonal. Specifically:

A o B o C o D
E o F o G o H
I o J o K o L
M o N o O o P
A o E o I o M
B o F o J o N
C o G o K o O
D o H o L o P
A o F o K o P
D o G o J o M

Note that the central operator o at position (4,4) is shared by both of the diagonal equations.

The question is: Can the letters A through P be replaced with all distinct numbers 1 through 16 in any order, and the placeholder operators o by + or * or = such that all 10 grid equations are satisfied? Operator precedence is the usual, no grouping is allowed.

Are there lots of solutions or none at all?

Answer 1

There are:

Many solutions

For example:

and the various rotations / reflections of the above. Also, swapping rows 1 and 2 and swapping rows 3 and 4 in the above solution produces a new solution (plus the associated rotations/reflections).

Answer 2

Here is another solution with a bit more variation in the equations.

 1 +  2 *  4 =  9
 * *  *    + +  +
13 + 11 = 16 +  8
 =    = =  =    =
 3 + 15 =  6 + 12
 + +  +    + +  +
10 *  7 = 14 *  5

This is the first of many that my program spat out, but it does not check that the centre symbol is the same for the two diagonal equations, so it is hard to tell how many solutions there are. Probably quite a lot.