[SPARK-56167][PS] Align astype with pandas 3 default string behavior

[ueshin]

What changes were proposed in this pull request?

This PR updates a few pandas-on-Spark astype paths to match pandas 3 behavior for the default string dtype.

In pandas 3, astype(str) returns the default string dtype and preserves missing values instead of converting them to string literals such as "NaN" or "<NA>". pandas-on-Spark still used the older behavior in a few localized conversion paths, including numeric, null, string, and boolean casts.

This PR makes three small changes in python/pyspark/pandas/data_type_ops/:

• update the shared string cast helper so astype(str) preserves missing values for pandas 3 string results
• align boolean-to-string casting with the same pandas 3 behavior, including the nullable metadata on the result field
• align string-to-bool casting for pandas 3 string-backed data with pandas' current astype(bool) result

Why are the changes needed?

Without this change, several pandas-on-Spark astype tests fail with pandas 3 because some conversion paths still follow the older string-casting behavior.

The failures came from two related mismatches:

• astype(str) converted missing values into string literals instead of preserving them as missing values
• some follow-up casts from pandas 3 string-backed data did not match pandas' current behavior

This patch fixes those localized mismatches while keeping the pandas 2 behavior unchanged.

Does this PR introduce any user-facing change?

Yes.

For pandas 3 users, pandas-on-Spark astype(str) now preserves missing values in the affected paths instead of converting them to string literals. This also fixes related behavior for boolean and string-backed casts that depend on pandas 3's default string behavior.

How was this patch tested?

The existing tests should pass.

Was this patch authored or co-authored using generative AI tooling?

Generated-by: Codex (GPT-5)

[ueshin]

cc @gaogaotiantian @HyukjinKwon @zhengruifeng