1. Can one still file new applications and maintain existing rights in all three countries?
Yes.
2. Are payments to Russia and Belarus allowed?
Yes, if they go through non-sanctioned banks. Western banks will not allow payments if they think they are in breach of sanctions.
3. Are PTOs of these countries still operational?
Yes. Even in Ukraine (Kyiv), the PTO did not interrupt operations, they continue mostly online. The Ukrainian Chamber of Commerce and Industry announced that martial law was introduced in Ukraine on February 24, 2022 for the term of 30 days (on March 15 it was extended for another 30 days), and that due to force majeure circumstances, missed business-related deadlines can be extended or restored. So far, the Ukrainian PTO has not issued any clear instructions in this regard.
We assume that any future regulations or directives will follow international practice, so it is safe to assume that any deadlines falling during the period when martial law is in force, shift to the date on which the martial law ends.
The European Patent Office has issued a notice concerning the application of Rule 134(5) EPC under the circumstances.