Do we know when a particle has a tangled pair? By looking at a particle alone, can we tell if it has a tangled pair somewhere in the universe?
Moreover when measuring the spin of a electron, we can't predict it, unless we know the spin value of its entangled pair. So how can we be certain its random and not the case where its value is located in its entangled pair but we just don't know who it is.
 A: 
So how can we be certain its random and not the case where its value is located in its entangled pair but we just don't know who it is.

Does it matter? Since we can only know about the (spin) state for certain after a spin measurement, prior to that it’s state is a general linear combination of up and down. In order to do any meaningful experiments we need to be able to prepare the system in any state of our choosing. Because only then do we know the dynamics of the system. 
Now take the case of $2N$entangled spins each put inside a box ($2N$ boxes in total), $N$ of which are in your lab and the other $N$ in a lab 1 light years away from you. Say the spin measurements were made in that lab 2 years ago. Those spin outcomes are random, half of them up and half of them down on average. Now if they know that they are entangled with your electrons, they know the outcome of your spin measurements. If they radio that information to you, you’ll know the outcome of your spin measurements. This is equivalent to having measured the spin since now you know the state. However if they don’t radio the information, the outcomes are unpredictable for you, thus random. 
In conclusion, the state of the system is dependent on how much you can possible know about the system (predictable probabilities of possible outcomes of measurements). If you assume a state and calculate probabilities which doesn’t match with the observed outcome, it means that we are using incomplete information which means our state is wrong. This is why we prepare the system in a state because then we know as much as we can know about the system and then carry out our experiments. 
A: 
By looking at a particle alone, can we tell if it has a tangled pair somewhere in the universe?

Short: No
Long: Let $\rho$ be the mixed state describing your particle in Hilbertspace $A$. Then there exits a purification $\psi$ in the Hilbertspace $A\otimes B$. Purification means that $\text{tr}_B \psi = \rho$.
Therefore $\psi$ and $\rho$ cannot be distinguished by just looking at Hilberspace A. Side remark $\psi$ is not unique, but there exits an infinite amount of purification's.
