In physics it is usually denote $dx^i$ by an "infinitesimal" variation of the variable $x^i$, and for example, the work could be written as: $$\delta W=\sum_i Y^i dx^i$$ (1) This implies that in sense $\delta W$ is a 1-form?
(2) How is the conceptual relation between an "infinitesimal" variation of $x^i$ and the differential of the coorditate $x^i$?
Many thanks!
To answer your first question, yes $dW$ is a $1$-form based on this definition $dW = \sum Y^i\,dx^i$. The idea is that $dW$ itself is something to be integrated to find the total work $W = \int dW = \int\sum Y^i\,dx^i$.
To understand the conceptual relation between an infinitesimal variation of $x^i$ and the $1$-form $dx^i$, consider as in single-variable integration the object $dx$ as "a small change in $x$". In terms of work and forces, $dx$ is a small displacement of the particle, say. Hence, $W = \int dW = \int F\,dx$.
In the multivariable setting the idea is the same in that each of the $dx^i$ accounts for "small displacements" in each direction so that $$ W = \int dW = \int \sum Y^i\,dx^i. $$