1. Are the liturgies of the Syriac Catholics and the Syriac Orthodox identical?
Welcome to the Forum! On paper yes, in praxis no.
2. Are the liturgies of the Syro-Malankara Church and the Syriac Orthodox Church identical (with the difference perhaps of Indian inculturation in the Syro-Malankara liturgy and the use of Malayalam)?
Yes. The differences are slightly in translation, and the Thubden (Dyptichs) commemorate the Pope. The Malankara Syriac Catholic ones are usually more abbreviated (meaning, closer to the oldest forms). The Orthodox Churches added some of their saints to their Thubden, as well as use the extended Thubdens regularly.
3. In regards to the Divine Office (Sh'himo), is the Office for the clergy and the laity identical? Or is the version for the laity abbreviated?
There are no lay versions, they use the same as priests, if possible, minus incense. However, there are no official "complete" Sh'himo. They vary by region. Mardin, TurAbdin, Damascus, Pampakuda (monastery), and some Bishop's editions have slight variations. The extended Divine Office can only be used in a monastery due to it's length - the rest are all abbreviated to some extent.
4. In regards to liturgical music for either the Divine Liturgy or the Divine Office, I understand that the foremost source is the Beth Gazo. Are its melodies standard? Or does every parish or region or country have its own melodies ?
The Beth Gazo is a standard, there are regional differences. Indian versions also vary regionally.
The Syriac Catholic Liturgy's norm is to be facing East, but facing the people is certainly popular - even the Patriarch and bishops do it. It's much, much rarer among Syriac Orthodox - although not unheard of.