So you have a lot of questions, which is understandable because this is all fairly complex and confusing. I think the first thing you should probably know is that DirectX shader programs generally only work in terms of 32-bit datatypes. So when you write HLSL code to add two values or transform a vector by matrix, the actual ALU operations are mostly going to be working with 32-bit floating point values or 32-bit integers. These are the only data formats that are guaranteed to be supported by the hardware in your shader programs, and most games/programs out there work exclusively with 32-bit operations. There are ways to work with operations that run at lower-than-32-bit precision, as well as the "double" type that can be used to access double-precision operations. However both of these are optional, and are only supported by certain GPU's. For most of the DX10 and DX11 era desktop GPU's only supported 32-bit operations, but recent AMD and Nvidia GPU's have started adding support for fp16 operations. Support for double-precision can be rather patchy, since it's generally only intended for high-end compute usage rather than 3D graphics.

So with that said, the other important thing you should know is that there's a difference between how your data is stored in buffers or textures, vs. how it's operated on in your shader programs. Your shaders have the ability to read or write data through various paths depending on the type of shader, and those paths can possibly support automatic conversion from its format in memory to the format that the shader program can consume it in. The classic case is textures: textures are almost never stored as a set of full 32-bit floats, and are instead typically stored as a fixed-point format like R8G8B8A8_UNORM. In order for a shader to sample a program, the texture has to specify that its data is laid out according to one of the supported DXGI format types from the DXGI_FORMAT enum. This is key, because these formats specify not just how the data is stored in memory, but also how the data is interpreted by a shader program. If you go to the docs for DXGI_FORMAT and scroll down to the section entitled "Format Modifiers", you'll see what I'm talking about. If we go to the commonly-used UNORM format modifer, you'll see this:

Unsigned normalized integer; which is interpreted in a resource as an unsigned integer, and is interpreted in a shader as an unsigned normalized floating-point value in the range [0, 1]. All 0's maps to 0.0f, and all 1's maps to 1.0f. A sequence of evenly spaced floating-point values from 0.0f to 1.0f are represented. For instance, a 2-bit UNORM represents 0.0f, 1/3, 2/3, and 1.0f.

The part that I bolded for emphasis is the important part: it's basically saying that the shader will see a floating point value in the range [0, 1], not an integer value. This lets you write shader code like this:

Texture2D<float4> ColorTexture;
SamplerState ColorSampler;

cbuffer Constants 
{
    float3 LightDir;
    float3 LightColor;
}

float4 PSMain(in float3 uv : UV, in float3 normal : NORMAL) : SV_Target0
{
    float3 surfaceColor = ColorTexture.Sample(ColorSampler, uv).xyz;
    float3 lighting = surfaceColor * saturate(dot(normal, LightDir)) * LightColor;
    return float4(lighting, 1.0f);
}

So the shader is just working entirely in float's, and doesn't care about the data format of the texture. The texture might be R8G8B8A8_UNORM, it might by R16G16B16A16_FLOAT, or it might be BC1_UNORM. It doesn't really matter to the shader, as long as the format is ultimately interpreted as a float. Hence you don't see any unpacking or conversion code here, we just go right to multiplying the texture sample with the lighting. Generally the only time you do care is if you use a UINT or SINT format, since those require decorating  the texture with <uint> or <int>. In those cases the integer values will be case to a 32-bit int type on read, which lets the shader work with them using 32-bit operations.

On a similar note, we can have data conversion happening when the pixel shader's output value gets written to a render target texture. The pixel shader outputs a float4, but the render target might be using R16G16B16A16_FLOAT, which is a common format for HDR rendering. If that's the case, the pixel shader output value will automatically get converted to that format on write, again without the shader really knowing or caring.

The same concepts extend to a few other places in the pipeline. Shaders can read from the "Buffer<>" type in HLSL, which maps to a shader resource view with D3D11_SRV_DIMENSION_BUFFER. This kind of buffer also uses a DXGI_FORMAT, and supports automatic conversion from the packed format to 32-bit values. RWTexture and RWBuffer UAV's can also perform format conversion on write, and possibly also on load depending on the GPU and OS that you're using.

The other place where format conversion is commonly uses is the input assembler (IA), which is responsible for reading from your vertex and index buffer and passing the vertex data to your vertex shader. For the formats that the IA supports, conversion works the same way it does for textures: transparently. Your HLSL code just needs to declare the input variable according to the way that the format is interpreted as specified in the docs, which means float for FLOAT/UNORM/SNORM, int for SINT, and uint for UINT formats. So if you have this as your VS input struct:
 

struct VertexInput
{
    float3 pos; 
    float2 tex;
    float3 normal;
};

Those 3 attributes could be using any of the FLOAT, UNORM, or SNORM formats in the vertex stream. You don't need to change your shader declaration if you change the vertex packing, as long as you're continuing to use formats with one of those 3 modifiers. This means that you can totally use lower-than-32-bit packing for your vertex attributes if you want to save on memory and bandwidth, and you absolutely should do this! You just have to make sure that the precision is adequate for your use cases. 

As for how the conversion happens, it's really up to the hardware. Texture format conversion is generally handled by dedicated hardware in the texture units, which is necessary because unpacking (and possibly decompression from a BC format) needs to happen before filtering can occur. on the flip side, some hardware (particularly mobile hardware) will patch code into the pixel shader to handle conversion from 32-bit floats to the render target format. For the IA, it depends. Some GPU's still have dedicated hardware and caches for fetching from vertex buffers. Some don't have any hardware for this, and instead will patch in a pre-amble to your vertex shader that reads and unpacks the vertex data. But even in those cases the decode cost tends to be pretty small compared to the cost of additional memory access, so tightly-packed data is most likely still going to be a win.

StructuredBuffer and ByteAddressBuffer have no format conversion applied to them on read. If you would like to do any packing of lower-precision values in there, you have to unpack then yourself in the shader program. Here's an example of what that might look like:
 

struct Vertex
{
    float3 Position;    // 3x32bit
    uint UV;            // 2x16bit UNORM
    uint2 Normal        // 4x16bit SNORM
    uint2 Color         // 4x16bit FLOAT
};

StructuredBuffer<Vertex> Vertices;

float U16NToF32(in uint val) 
{
    return saturate(val / 65535.0f);
}

float2 S16NToF32_2(in uint val) 
{
    int2 signextended;
    signextended.x = (int)(val << 16) >> 16;
    signextended.y = (int)(val & 0xFFFF0000) >> 16;
    return max(float2(signextended) / 32767.0f, -1.0f);
}

VSOutput VSMain(in uint VertexID SV_VertexID)
{
    Vertex vertex = Vertices[VertexID];    
    float3 position = vertex.Position; 

    // Unpack by converting to full 32-bit values
    float2 uv = float2(U16NToF32(vertex.uv & 0xFFFF), U16NToF32(vertex.uv >> 16));
    float4 normal = float4(S16NToF32_2(vertex.Normal.x), S16NToF32_2(vertex.Normal.y));
    float4 Color = float4(f16tof32(vertex.Color.x), f16tof32(vertex.Color.x >> 16),
                          f16tof32(vertex.Color.y), f16tof32(vertex.Color.y >> 16));
 
    ...                          
}

The same goes for UAV's. RWStructuredBuffer an RWByteAddressBuffer have no format conversion applied on write, but RWBuffer<> and RWTexture do.