$\phi^{4}$ theory
I think, the sentence you are citing is meant to give a rough idea about the intuitive meaning of the $\phi^4$ term, not to be verified computationally.
The solution of the free theory (without the interaction term) is what defines things like "particles" and propagators. $a^\dagger$ as defined in the free theory's expansion creates a free particle. Usually, if you consider a scattering process, before the actual scattering occurs the particles are considered free.
They are still there. Usually you do the expansion not in the Lagrangian but when you write down a scattering amplitude. Then contractions of creator and annihilator operators will lead to commutators which will produce delta distributions which consume the momentum integrals.
Not sure if I understand the question correctly but if you have a look at the Feynman rules for the different terms in the Lagrangian, you will see that a $\phi^4$ term corresponds to a vertex with four legs while the free Lagrangian (which is quadratic in the field) corresponds to a propagator with two legs. So the number of legs in a Feynman rule corresponds to the power of the field in the corresponding term.
Let's look at Tong's comment in context, by which I mean start reading Chapter 3 from the start.
He's discussing interacting fields, and begins by writing a Lagrangian that features arbitrarily high powers of $\phi$. However, since in a $4$-dimensional spacetime the energy dimension of $\lambda_n$ is $4-n$, it follows that $\lambda_n$ with $n>4$ are supressed at low energies, viz. $\lambda_n=g_n\Lambda^{4-n}$ with $\Lambda$ a new-physics energy scale. Thus at an energy $E\ll \Lambda$ we get $\lambda_n\propto \left(\frac{E}{\Lambda}\right)^{n-4}$, which is small for $n>4$. This observation motivates a classification of coefficients as relevant, marginal or irrelevant if their energy dimensions are respectively positive, zero or negative. (The term "relevant" means "dominant at the scales we've probed"; "marginal" means "scale-independent, thus dominant at neither high nor low energies".) Tong notes that the fact only finitely many terms are relevant or marginal simplifies QFT.
Then we get to the comment you asked about, in which he discusses how several theories ($\phi^4$ being the first) behave under small perturbations. He's thrown away irrelevant couplings, but $\phi^4$ has been retained, and is the only such term that's not also present in the case of free fields. We recover the free-field case as $\lambda_4\to 0$, so the $\phi^4$ theory with small $\lambda_4$ is a perturbation around the theory of a free field, making your integral representation of $\phi$ approximately valid. The exact expression for $\phi$ therefore adds on some $\lambda_4$-dependent terms that, in comparison to the original integral, are quite small. It is therefore still reasonable to think of what happens to ladder-operator monomials' matrix elements.