A Hurewicz fibration $E\rightarrow B$ is a map so that for all maps $X\rightarrow E$ and $X\times I\rightarrow B$ making the obvious square commute, there is a lifting $X\times I\rightarrow E$ so that everything commutes. This is known as the homotopy extension property for $X$. A Serre fibration is a map that satisfies the homotopy extension property for cubes.
My question is: first, given a Hurewicz/Serre fibration, for what pairs $A\rightarrow X$ mapping into $E\rightarrow B$ can we find a map $X\rightarrow E$ making the diagram commute? For example, this is true for $X\rightarrow X\times I$ for Hurewicz fibrations; I would like to know a complete characterization for such pairs.
Also, is there a name for maps $E\rightarrow B$ satisfying this sort of lifting property for a specific pair $A\rightarrow X$? I would guess it would be named something like "homotopy lifting property relative to $A\rightarrow X$" but I have failed to find something like this in my searches.
In addition, I would be interested in the dual questions for cofibrations. These questions feel simple enough that there should be references for them, but I haven't found them.
The fibrations in any model catetgory are characterised by their lifting property with respect to a particular class of maps. In detail:
Below I will call any morphism which is both a cofibration and a (weak) equivalence an acyclic cofibration. Similarly for fibrations.
Now, let $Top$ denote either the category of all topological spaces or, say, all compactly-generated weak Hausdorff spaces. Then $Top$ supports two relevant model structures:
The first has homotopy equivalences as weak equivalences, Hurewicz fibrations as fibrations, and closed cofibrations as cofibrations. Call this the Strom model structre.
The second has weak homotopy equivalences as weak equivalences, Serre fibrations as fibrations, and the cofibrations are the maps which have the left lifting property with respect to all acyclic Serre fibrations. Call this the Serre model structure.
So this answers your first question. Working with the Strom model structure we see that the Hurewicz fibration are exactly the maps which have the right lifting property with respect to all maps $i:A\xrightarrow\simeq X$ which are both homotopy equivalences and closed cofibrations. Note that a map which is both a cofibration and a homotopy equivalence is a strong deformation retract, although the converse is not true. Note also that any cofibration between weak Hausdorff spaces is closed, although there do exist non-closed cofibrations (which are even homotopy equivalences) in the larger category.
The acyclic cofibrations in the Serre model structure have a more awkward description. They are maps which are retracts of transfinite compositions of spaces constructed as pushouts of sums of inclusions of the form $D^n\hookrightarrow D^n\times I$. One of the benefits of Serre model is that it is cofibrantly generated, and you'll find a slightly more accurate description of these maps by looking up this term.
Turning to your second question. In general it's said that a map $p:E\rightarrow B$ has the homotopy lifting property (HLP) with respect to a space $X$ if for any map $f:X\rightarrow E$ and any homotopy $H_t:X\times I\rightarrow B$ with $H_0=pf$, there exists a homotopy $G:X\times I\rightarrow E$ with $G_0=f$ and $pG_t=H_t$.
Then the Hurewicz fibrations are those maps which have the homotopy lifting property with respect to all spaces, and the Serre fibrations are those maps which have the homotopy lifting property with respect to all discs $D^n$, $n\geq0$. Note that for any $X$ the inclusion $in_0:X\hookrightarrow X\times I$ is an acyclic cofibration in both model structures.
Useful examples abound: A map has the HLP with respect to the one-point space if and only if it has path-lifing. A locally trivial map has the HLP with respect to all paracompact $T_2$ spaces. A map has the HLP with respect to all CW complexes if and only if it is a Serre fibration.
The last thing to adress is that yes, all of this is dualisable. This is really the power of realising the model category framework: the existence of the Strom and Serre model structures really should be celebrated facts. There is an equally valid description of the cofibrations, acyclic cofibrations and acyclic fibrations in any model category.
An accessible reference for model categories would have to be Mark Hovey's book Model Categories. If you are interested in seeing the full statement with respect to the Strom model structure on topological spaces, then let me direct you to Jeff Strom's book Modern Classical Homotopy Theory $\S$5.6.4.