I am a permanent resident based on asylum, a green card holder, citizenship is Russia. I was advised not to use my national passport with the US immigration as it may signal my willingness to accept benefits of the country from which I was granted asylum in the US. I currently possess a valid Russian passport.
When I want to go travel to a certain country that has a visa waiver for Russian citizens, would it be okay to present the Travel Document on the way out of the US and enter the destination country on the national passport?
If I decide to travel to a country for which I need a visa as a permanent resident asylee, would it be okay to travel to neighboring countries presenting the national passport to avoid receiving visas as a Russian citizen and subsequently return to the US from the country of the initial arrival presenting the Travel Document upon arrival to the US?
Details for Q2. The nature of the question is about a potential issue of not having all of the arrival/departure stamps in the same document. To clarify, I do not intend to travel to Russia. To paint the picture better, let's say my itinerary is USA, Thailand, Vietnam, Indonesia, Thailand, USA. A visa would be obtained only for Thailand and placed in the Travel Document.
Note: this is related to I have two passports/nationalities. How do I use them when I travel? but not the same as in this case there is one passport and one nationality.