Punjabi

Yes. Punjabi has three tones (high-falling, low-rising, level), making it the rare Indo-European language with phonemic tone. The tones developed from a historical loss of voiced aspirated consonants (gh, jh, dh, etc.), which left tonal traces on the surrounding vowels. The same word with different tones can mean entirely different things.

Religious and political history. Indian Punjab uses Gurmukhi, the script the Sikh Gurus standardized in the 16th century. Pakistani Punjab uses Shahmukhi, a variant of the Perso-Arabic script used for Urdu. The two scripts represent the same language, but a literate Punjabi speaker on one side of the border usually can't read material from the other without learning the second script.

The Punjab region of South Asia, divided in 1947 between India (about 33 million speakers) and Pakistan (about 80+ million). Pakistani Punjab is the most populous province in Pakistan, but Urdu is the language of education there, so written Punjabi is less developed than its spoken use suggests. Punjabi is also one of Canada's most-spoken non-official languages.

Same Indo-Aryan branch, similar grammar (postpositions, SOV, gender, split-ergative past), but Punjabi adds tone and uses different vocabulary at higher registers. Pakistani Punjabi shares much of its everyday vocabulary with Urdu; Indian Punjabi shares more with Hindi. The three are not mutually intelligible without exposure.

Yes, two: masculine and feminine. Verbs, adjectives, and some postpositions agree with the noun. The same split-ergative past tense found in Hindi, Urdu, and Marathi shows up in Punjabi too: with transitive past verbs, the verb agrees with the object, and the subject takes the postposition ne.