To fix the notation, a spectrum is a sequence of spaces indexed over $\mathbb{N}$ (!) with structure maps as usual.
My first questions is: Does the suspension spectrum functor embed the category of spaces into spectra? It is clearly a faithful functor and injective on objects but I doubt that it is full.
There is a unstable model structure on spectra by defining weak equivalences and fibrations degreewise.
My second question is: Does the suspension spectrum functor embed the homotopy category of spaces into the unstable homotopy category of spectra?
Define two endofunctors $s$ and $t$ on the category of spectra by setting $sX_n=X_{n+1}$ and $tX_n=X_{n-1}$ where $X_{-1}=*$. Then $t$ is left adjoint to $s$ and this is a Quillen adjunction with respect to the unstable model structure.
My third question is: Is it true that $s$ is left adjoint to $t$, too? If yes, is this also a Quillen adjunction? I have checked it and it seems to be right but somehow I feel not well about it.