You have a few confused ideas here- which is quite normal, because it is a confusing subject that is often not described very well. Instead of addressing them all specifically, I will answer the central question:
Entropy is objective.
It is true that one must specify exactly what degrees of freedom are in and out of your system to say what the entropy is. However, this is true for any other physical property, like energy, as well, so entropy is as objective as any other physical quantity.
Since I'm a physicist I'll talk about particles, but anything I'm saying can be adapted to cards or whatever system you're thinking of.
It's pretty obvious that the energy in a system depends on how you define the system. For example, adding particles into your system increases the total amount of energy you measure. Less obviously, you could choose to include or exclude certain degrees of freedom within each particle. For example, every particle with mass has associated energy given by $E=mc^2$, but if one doesn't know this or doesn't want to worry about it one could only consider the kinetic energy of the particles and ignore this degree of freedom.
Entropy is much the same*. Once you specify exactly which particles and which degrees of freedom within each particle you are keeping track of, it is uniquely specified. Choosing to ask more or less precise questions- such as asking exactly where a particle is in a bottle, versus just asking whether it is on the left or right side- amounts to ignoring more or fewer (or possibly zero) degrees of freedom.
There are two important differences between entropy and energy that I should note.
The first is that if you do include all the degrees of freedom of an isolated system, the entropy you calculate is always zero**. Since entropy is something like uncertainty, this is a statement that you have all of the possible information about the system. Once you start to leave parts out, the entropy becomes greater than zero.
The second is that, roughly speaking, in practice one can often ignore many degrees of freedom and still count the same total energy, while entropy is much more sensitive to this. This is because entropy is sensitive to correlations between particles, while energy is not.
This is one way of describing the phenomenon of quantum entanglement.
See also my answer to a related, slightly more technical question: What is the entropy of a pure state?
*Technical qualification 1: I will have in mind the Von Neumann entropy, defined as $S=tr(\rho \log (\rho))$, with $\rho$ as the density matrix of the system. However, I believe that my statements apply to any other conventional definition of entropy in a physics context.
**Technical qualification 2: only if the system started in a pure quantum state.
Edit (07/15/15): Since this is clearly a contentious claim, I will give a toy example, at a higher technical level than my main answer. I will compare two example systems in which degrees of freedom are traced out.
First, an example of course graining in position. Let's say I have a particle that can be in one of four positions in a box:
__________________
| | |
| A1 | B1 |
| | |
| A2 | B2 |
|________|________|
There are sides A and B, and within each side are two sites.
The initial state is:
$\rho=\frac{1}{\sqrt{2}}(|A1\rangle + |B2\rangle)\otimes \frac{1}{\sqrt{2}}(\langle A1| + \langle B2|) $
Or in matrix form:
$\rho=\frac{1}{2}\left[
\begin{array}{cccc}
1&0&0&1 \\
0&0&0&0 \\
0&0&0&0 \\
1&0&0&1
\end{array} \right]$
The columns and rows in this matrix are labeled like $|A1\rangle,|A2\rangle,|B1\rangle,|B2\rangle$. The entropy of this matrix, like any pure state, is 0.
Now, say we are only interested in whether the particle is on side A or B. We coarse grain by tracing out the extra degree of freedom:
$\rho_{red}=\langle 1|\rho|1\rangle+ \langle 2|\rho|2\rangle$
or
$\rho_{red}=\frac{1}{2}\left[
\begin{array}{cc}
1&0 \\
0&1
\end{array} \right]$.
The columns and rows in this matrix are labeled like $|A\rangle,|B\rangle$.
This is now a reduced density matrix for only the A/B degree of freedom. Since we started with an entangled state, the entropy is also now nonzero.
Now I'll turn to an apparently unrelated system, of two spin-1/2 atoms. The initial state is:
$\rho=\frac{1}{\sqrt{2}}(|\uparrow_1 \uparrow_2 \rangle + |\downarrow_1 \downarrow_2\rangle)\otimes \frac{1}{\sqrt{2}}(\langle \uparrow_1 \uparrow_2 | + \langle \downarrow_1 \downarrow_2|) $
Or in matrix form:
$\rho=\frac{1}{2}\left[
\begin{array}{cccc}
1&0&0&1 \\
0&0&0&0 \\
0&0&0&0 \\
1&0&0&1
\end{array} \right]$
The columns and rows in this matrix are labeled like $|\uparrow_1 \uparrow_2 \rangle,|\uparrow_1 \downarrow_2 \rangle,|\downarrow_1 \uparrow_2 \rangle,|\downarrow_1 \downarrow_2 \rangle$. We again start with a zero entropy pure state.
Now let's say we want to only consider one of the particles. We must trace out the degree of freedom associated with the other particle. But this goes through exactly in analogy to the previous example, and we end up with
$\rho_{red}=\frac{1}{2}\left[
\begin{array}{cc}
1&0 \\
0&1
\end{array} \right]$.
The columns and rows in this matrix are labeled like $|\uparrow_1\rangle,|\downarrow_1\rangle$. At the level of the state structure of these two systems (which is all that determines entropy), the mapping between them is exact, and the entropy of the reduced system is again nonzero.
With all this setup, my point is simple. In my second example, I would be greatly surprised if anyone claimed that the entropy is arbitrary. You get two different results for the entropy by asking two different questions: what is the entropy of one particle versus what is the entropy of the full two-particle system. The fact that these have different answers is no deeper than the fact that you get different answers if you ask what the energy of one particle is versus the energy of two.
But exactly the same claim applies to the first system. In that case the degree of freedom you threw away didn't happen to be associated with a single particle, but it is still true that you would only get different answers for the entropy by asking different questions, and in particular by asking about different parts of the total system. There is no good reason to privilege degrees of freedom that happen to be contained in individual particles over those that do not. Since the only "subjective" aspect of entropy is the trivial fact that you have to choose precisely what you want to know the entropy of, it should be considered as objective as any other physical property.
