Electronic – can we do one always inside another always for verilog

When you assign to a register in an edge-sensitive always block, you're defining a flip-flop. FPGAs do not have flip-flops that can trigger on both edges of a clock.

In order to do what you want, you are going to need to have two separate always blocks, one for each edge of the clock, and then figure out a way to combine the outputs of the two blocks without creating glitches.

For example, one always block could contain your programmable divider. Design it so that the output duty cycle is less than 50% when given an odd number. Use the second always block (on the other clock edge) to delay the output of the first block by 1/2 clock, then OR the two outputs together. Disable the output of the second block for even divider values.

Your code simulates two multiplexers. These are actually asynchronous components. The fact that Verilog requires data1_temp and data2_temp to be declared as reg's is a quirk of Verilog syntax and your choice of coding style, and doesn't mean these signals would be the outputs of storage elements in a physical implementation.

If you want to capture these values in actual registers, you need to add those explicitly:

reg [7:0] data1, data2;
always @(posedge someclock) begin
    data1 <= data1_tmp;
    data2 <= data2_tmp;
end

But I would like to know what this mini register file would be made of in hardware. Particularly, the 4x8 bit array consisting of k0,k1,k2,k3.

You haven't shown how these variables are assigned, so it's not possible to say how they are implemented. As your code showed, just declaring them as reg doesn't guarantee they are implemented with actual storage elements. If you assign them inside a block that begins always @(posedge clk) then very likely they are flip-flops, but there are ways you could code them that would make them synthesize differently.

I thought when it came to registers and arrays like this, you need a clock to read out data, like RAM?

You need a clock to update a (physical) register. You can read it out at any time. For example:

wire [8:0] sum;
assign sum = k0 + k1; 

is perfectly valid code. sum will change whenever any of its inputs changes. If k0 and k1 are the outputs of flip-flops, their values will only change when there is a clock edge.

For another example, you could equally well describe your multiplexers with code like this:

reg [7:0] k0, k1, k2, k3;
wire [7:0] data1_tmp;
reg [1:0] reg1;
// k<n> and reg1 are assigned elsewhere.
assign data1_tmp = (reg1 == 0) ? k0 :
                   (reg1 == 1) ? k1 :
                   (reg1 == 2) ? k2 : k3;

how do I read from this tag_array and do the comparison all within the same clock cycle?

Let me repeat a key point for emphasis: You need to use a clock to assign a new value to a register (an actual hardware register or group of flip-flops). It's output is available at any time.

RAMs are different and how you access the contents of a RAM will depend on details of the type of RAM you use.

I got confused because frankly I don't know enough about memory hardware and how that's possible.

Another key strategy: When you are learning digital logic, I recommend you learn about the physical hardware first, and then work out or study how to simulate it in HDL second. So first, learn what a physical flip-flop is, then learn the standard Verilog methods of describing a flip-flop. Especially if you are trying to write HDL for synthesis, trying to write good code before you learn the capabilities of the underlying hardware will lead you down a lot of dead-end paths.