I'll answer your second question first. To some degree, the ordering you choose will largely depend on whether you want to lead with examples or with full abstraction. As an example, you can introduce projective resolutions and the derived category using only facts about $\text{Ch}(\mathcal{A})$ and Ore's calculus of fractions (see Weibel's book for a treatment like this) or you can introduce model categories, prove their properties, prove that $\text{Ch}(\mathcal{A})$ admits a projective model structure using a small object argument, and arrive thus at a description of the derived category as a homotopy category.

Personally I think the second account would be unnecessarily convoluted and it would make more sense to introduce some homological algebra first, not least because that way you can introduce the projective model structure as an example of a model structure, projective resolution as an example of a cofibrant resolution, derived category as an example of a homotopy category et cetera. But both are available to you!

On the question of model categories and quasicategories: model categories can be viewed as "presentations" for quasicategories (see this nLab page for this perspective, and Appendices A.2 and A.3 of Lurie's Higher Topos Theory for a development of the theory of model categories with this explicit goal). Quasicategories have several advantages over model categories: for example, there is a quasicategory of functors from any quasicategory to another, whereas the analogous statement doesn't hold for model categories. Model structures are heavily involved in many of the foundational proofs regarding quasicategories, however, so there's no two ways of ordering these topics.

On your first question: personally I don't believe homological algebra is sufficient motivation for introducing either model categories or infinity-categories. As raised in the comments, the triangulated category $\mathcal{D}(\mathcal{A})$ doesn't allow functorial cones and this is annoying in some applications, but people mostly got on fine with applying homological algebra for decades before people started talking about dg- and quasicategories. A stronger order for your text, in my opinion, would be to introduce basic concepts from homological algebra, then use these as examples when you start talking about model categories and finally quasicategories.

On the question of a universal property for $\mathcal{D}(\mathcal{A})$ using infinity categories, you might find section 1.3.3 of Lurie's Higher Algebra helpful. Note, however, that $\mathcal{D}(\mathcal{A})$ certainly does have a universal property in ordinary 1-categorical language: it is the localization of $\text{Ch}(\mathcal{A})$ at the quasi-isomorphisms.

I'll answer your second question first. To some degree, the ordering you choose will largely depend on whether you want to lead with examples or with full abstraction. As an example, you can introduce projective resolutions and the derived category using only facts about $\text{Ch}(\mathcal{A})$ and Ore's calculus of fractions (see Weibel's book for a treatment like this) or you can introduce model categories, prove their properties, prove that $\text{Ch}(\mathcal{A})$ admits a projective model structure using a small object argument (see this nLab page for an outline of the argument), and arrive thus at a description of the derived category as a homotopy category.

Personally I think the second account would be unnecessarily convoluted and it would make more sense to introduce some homological algebra first, not least because that way you can introduce the projective model structure as an example of a model structure, projective resolution as an example of a cofibrant resolution, derived category as an example of a homotopy category et cetera; these concepts can be difficult to gain an intuition for without several examples! But both orderings are available to you.

On the question of model categories and quasicategories: model categories can be viewed as "presentations" for quasicategories (see this nLab page for this perspective, and Appendices A.2 and A.3 of Lurie's Higher Topos Theory for a development of the theory of model categories with this explicit goal). Quasicategories have several advantages over model categories: for example, there is a quasicategory of functors from any quasicategory to another, whereas the analogous statement doesn't hold for model categories. Model structures are heavily involved in many of the foundational proofs regarding quasicategories, however, so there's no two ways of ordering these topics.

On your first question: personally I don't believe homological algebra is sufficient motivation for introducing either model categories or infinity-categories. As raised in the comments, the triangulated category $\mathcal{D}(\mathcal{A})$ doesn't allow functorial cones and this is annoying in some applications, but people mostly got on fine with applying homological algebra for decades before people started talking about dg- and quasicategories. A stronger order for your text, in my opinion, would be to introduce basic concepts from homological algebra, then use these as examples when you start talking about model categories and finally quasicategories.

On the question of a universal property for $\mathcal{D}(\mathcal{A})$ using infinity categories, you might find section 1.3.3 of Lurie's Higher Algebra helpful. Note, however, that $\mathcal{D}(\mathcal{A})$ certainly does have a universal property in ordinary 1-categorical language: it is the localization of $\text{Ch}(\mathcal{A})$ at the quasi-isomorphisms.