Let $Y$ be a topological space.
Let $f:\mathbb{S}^1=\partial\mathbb{D}^2\rightarrow Y$ be a continuous map.
By attaching $2$-cell to $Y$ we mean the space $Y\bigsqcup \mathbb{D}^2$ under the identification that $x\in \mathbb{S}^1$ is identified with its image $f(x)$ in $Y$.
Let us denote the quotient space $(Y\bigsqcup \mathbb{D}^2)/\sim$ by $X$.
Question : What all properties are preserved under attaching a $2$-cell?
I am interested (but not limited to) Hausdorff, locally Hausdorff, regular, normal, compact, paracompact, contractible, locally contractible.
Your question is a little broad. Some results are sketched in Laz's comments. This community wiki intends to collect known facts. The space $X$ is a special case of a relative CW-complex. The Appendix of [1] easily generalizes to give proofs of a lot of results. Section 1.8. of [2] contains a number of very general results on adjunction spaces.
Here are some answers based on these sources.
1) Separation axioms.
$T_1$ : Yes.
Hausdorff: Yes.
Regular: Yes.
Normal: Yes.
2) Compact.
Yes because you have a continuous surjection from the compact space $Y \bigsqcup \mathbb{D}^2$ onto $X$.
3) Contractible.
No. Take $Y$ a single point space. Then $X$ is homeomorphioc to the $2$-sphere.
4) Locally contractible.
Yes. Modify the proof of Proposition A.4 in [1].
[1] Allen Hatcher, Algebraic topology
[2] Tammo tom Dieck, General Topology, https://www.uni-math.gwdg.de/tammo/GT01.pdf