Is everything entangled? In certain experiments we can entangle two photons, two electrons, and even a photon and an electron. It's also possible to exchange one of the particles in an entangled pair with a different particle.
In light of this, is it likely that everything is entangled, we just don't know what with except when we run these experiments?
 A: It depends on your definition of the word "entangled".
If you take the meaning "interweaved" , quantum mechanical solutions, the wave functions $Ψ$ of a system or an interaction , entangle all the particles involved because conservation laws impose definite values when a measurement is made, for all the variables and quantum numbers involved. That is the way the particle interactions have been studied  building up the standard model of particle physics . Event per event is recorded with the measurements of the outgoing particles defining the interactions at the vertex, getting the crossections and decay amplitudes for the particles involved.
If you take the meaning to "entangled" given by the quantum information :

Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated.
This leads to correlations between observable physical properties of the systems
For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be spin-up, the other one will always be observed to be spin-down and vice versa, this despite the fact that it is impossible to predict, according to quantum mechanics, which set of measurements will be observed.

The italics mine  to show where the use of the term differs from the experiments in particles in high energy physics and quantum computing.
The experiments at the LHC use energy momentum and angular momentum conservation of the entangled particles  to record events as they happen, building up distributions to be compared with predictions of theory.
Quantum computing tries to set up systems where they can be individually used to have information about one part of the system even though it is very much separated from the whole.
So to your question:

In certain experiments we can entangle two photons, two electrons, and even a photon and an electron. It's also possible to exchange one of the particles in an entangled pair with a different particle.

Here specific quantum numbers are used to check on the entanglement .

In light of this, is it likely that everything is entangled, we just don't know what with except when we run these experiments?

In principle yes, theoretically one wave function can describe the universe, with the general quantum entanglement definition.
In quantum computing one is interested in specific quantum numbers, as spins that have an easy value, up or down,  + or -, that are easy to measure and use.
My example of entanglement of this type in particle physics is the $π^0$ particle.It decays into two $γ$. The $π^0$ has spin zero, so if one measures the spin of one gamma, one knows the spin of the other gamma no matter how far it has gone, from conservation of angular momentum. ( It is impossible to generate a $π^0$ on purpose, as it is controlled by the probabilistic amplitude of the interaction that created it.)
