Is the innate immune system capable of identifying all foreign antigens?
TL;DR the huge block of text:

The innate immune system only recognizes certain antigens, called PAMPs.
The activate immune system usually requires the immune system, but it can get around this in some cases.


You're on the right track.
Pathogen-associated molecular patterns (PAMPs)
Innate immune cells such as macrophages and denritic cells (DCs) only recognize certain pathogen-associated molecular patterns (PAMPs), and this occurs primarily through membrane-bound receptors called Toll-Like Receptors (TLRs). Diversity of PAMPs is extremely limited compared to the extraordinary diversity in recognition by the active immune system. However, they include many molecules and classes of molecules that are essential for certain pathogens, such as flagellin, lipopolysaccharide, and viral double-stranded RNA.
To answer some of your question right away, the immune system can't adapt, but it can recognize antigens that the body hasn't seen before, because the set of antigens it can recognize (the PAMPs) are dictated by the genes associated with Toll-Like Receptors and other &quot;Pattern-Recognition Receptors&quot;.
Exception 1: T cell anergy
The major link between the innate and active immune systems that you mentioned is through the presentation of antigens to T helper cells in lymph nodes by DCs: DCs present antigens (digested parts of pathogens) to T helper cells through a process called antigen cross-presentation.
When this happens, the T helper cell is activated through a series of chemical exchanges between the DC and the T helper cell, with two that are important to mention here: 1) the T cell recognizes the antigen–MHC class II complex on the DC (called T cell stimulation), and 2) the T cell receives a signal (called co-stimulation). The first signal only requires cross-presentation and recognition by a T cell. But for the DC to release the co-stimulatory signal, Toll-Like Receptors (TLRs) on its surface need to have recognized a PAMP.
With these interactions (plus some more), the T cell will begin proliferating. However, immunology is very nuanced; you can probably find exceptions to every sentence I just wrote (I see a lot even while writing this). It turns out that a T cell can still be activated even without co-stimulation if its own stimulation is strong enough. This occurs through a process called T cell anergy. Activation of an anergic T cell—that is, one that was activated but did not receive a co-stimulation signal—has to do with the autocrine and paracrine nature of a signal called IL-2, which T cells normally produce only small amounts of but produce much more when fully stimulated.
Exception 2: B cells T-independent activation
But T cells are only half of the active immune system. It turns out that B cell activation follows similar rules, though not through DCs. B cell activation usually occurs with help from activated T helper cells (only one that recognizes the antigen the B cell is specific for). However, B cells can become activated without T cell help in two situations: an antigen binds to the B cell receptor and to a TLR, or 2) the B cell receptors become saturated with highly repetitive epitopes. Situation #2 technically occurs without help of the innate immune system.
So you're mostly correct that the active immune system requires innate immune system help, except that there are these two exceptions. And since immunology has so many exceptions, there are likely to be more.