For instance take the definition of equalizer, it's difficult for me to remember which direction the existing unique arrow goes, so it's all right usually to reverse it, correct?
So if $f,g : A \to B$, $k = \ker(f,g), \ k : C \to A$ then for any $f\circ l = g\circ l$ with $l : D \to A$ there is a unique arrow from $n : D \to C$ such that $k \circ n = l$. I can never remember if it's from the hypothesized other object or the object under definition when talking about a UMP. So is it okay to reverse this arrow (there exists $n : C \to D$ such that $k = l\circ n$) usually when refering to a definition later?
No. Not even remotely. For example, if this were the case you would not be able to differentiate between a terminal and an initial object.
Your issue with remembering which way the arrows go is one of the reasons I prefer the representability/isomorphism of hom-sets approach over universal arrows. For example, presenting $F\dashv U$ as $\mathsf{Hom}(FA,X)\cong\mathsf{Hom}(A,UX)$, it is pretty easy to keep track of things. Even when you don't have an adjunction, a universal property can at least be represented via representability, e.g. $\mathsf{Hom}(A+B,X)\cong\mathsf{Hom}(A,X)\times\mathsf{Hom}(B,X)$ natural in $X$ but not necessarily in $A$ and $B$. (If it is natural in $A$ and $B$, we quickly get an adjoint situation by viewing the right hand side as the hom-set of the product category.)
The world of universal properties is then split into limits/right adjoints/mapping-in properties and colimits/left adjoints/mapping-out properties. In many cases, very basic but important results like the continuity of hom-functors and adjoints further help remembering which way things go, e.g. the property above about $+$ is an example of continuity of the hom-functor.
For your equalizer example, it's a limit so we want a mapping-in property, namely $\mathsf{Hom}(A,\mathsf{ker}(f,g))\cong\{h\in\mathsf{Hom}(A,X)\mid f\circ h = g\circ h\}$ where $f,g : X \to Y$. This characterization comes straight from continuity of the hom-functor and the definition of an equalizer in $\mathbf{Set}$. (Well, technically, in a functor category, but those are calculated pointwise if the codomain is complete.) This description makes it clear that the induced arrows are arrows mapping in to $\mathsf{ker}(f,g)$. And just to reiterate, if you then wondered whether it should be $\mathsf{Hom}(A,\mathsf{ker}(f,g))$ or $\mathsf{Hom}(\mathsf{ker}(f,g),A)$, a variety of factors would remind you. First, since it's a limit it should be on the right. Second, if you try to express right-hand side of the natural isomorphism from before it will be a functor with the wrong "variance", i.e. the functors will be in different categories: $\mathsf{Hom}(-,\mathsf{ker}(f,g)) : \mathcal{C}^{op}\to\mathbf{Set}$ while $\mathsf{Hom}(\mathsf{ker}(f,g),-):\mathcal{C}\to\mathbf{Set}$.