Learn Polish: Zawarłszy sojusz z sąsiadem, królestwo odzyskało spokój. - Having concluded an alliance with the neighbor, the kingdom regained peace.

It is the adverbial anterior participle of zawrzeć (to conclude). To form it, you take the 3rd person singular masculine past tense form, which is zawarł. Because this base ends in a consonant, you add the suffix -szy. If the past tense form had ended in a vowel, you would add -wszy instead.

Since the subject doing the action is królestwo (a neuter noun), why doesn't zawarłszy have a neuter ending?

Adverbial participles in Polish are indeclinable. This means they never change their form to match the gender, number, or case of the subject. The form zawarłszy stays exactly the same whether the subject is masculine, feminine, neuter, or plural.

Could we use this same participle structure if a king made the alliance, but the kingdom regained peace?

No, you can only use this type of participle if the subject of both actions is exactly the same. Here, królestwo is the subject that both made the alliance and regained peace. If the subjects were different, you would have to use a full time clause, such as Po tym jak król zawarł sojusz, królestwo odzyskało spokój.

Why is the verb zawrzeć used here with sojusz?

The phrase zawrzeć sojusz is the standard Polish collocation for making or concluding an alliance. You will frequently see the verb zawrzeć used for formal binding agreements, such as zawrzeć umowę (to sign a contract), zawrzeć pokój (to make peace), or zawrzeć małżeństwo (to get married).