Learn Arabic (Modern Standard): صار الفرش قديما.

This happens because of the verb صار (to become). صار belongs to a famous family of verbs called the "Sisters of Kana" (أخوات كان). These verbs make their predicate (the descriptive part of the sentence) take the accusative case. Because "old" is the predicate here, it takes a double fatha ending (-an). In Arabic spelling, this requires an extra alif to act as a "chair" for the double fatha, making it قديما (qadīman).

I usually use أصبح for "to become". Are أصبح and صار interchangeable here?

They are very similar and both follow the exact same grammar rules. However, صار strongly emphasizes a transformation or a transition from one state to another (in this case, transitioning from new to old). أصبح is a bit more general and is the most common default word for "to become" in Modern Standard Arabic, but either would be perfectly understood here.

I know the word فراش (firāsh) means "bed". Is الفرش (al-farsh) related to it?

Yes, they share the same root (ف-ر-ش), which carries the core meaning of "spreading out". While فراش specifically means a bed or mattress, الفرش is a broader noun. It refers to things that are spread out to furnish a space, such as carpets, mats, cushions, or in this context, the upholstery of furniture.

How can I tell if الفرش is a masculine or feminine word based on this sentence?