Questions about the titles "count" and "prince" in 19th century Russia

by White_Heart

I am currently reading War and Peace by Leo Tolstoy, and I have some questions about the titles count and prince, which appear almost on every page of this novel.

Which of these titles has a higher ranking within the aristocracy? Are they degrees on the same ladder, or are they in completely unrelated categories? Because in the novel, Prince Bolkonski considered his son marrying a countess unfavorable, while Prince Vasili basically arranged his daughter's marriage to Count Bezukhov because he thought it was advantageous.
Do princes and counts have a direct familial relationship to the Tsar? Or are they from different, unrelated families?
Is it possible to acquire those titles? Like when a person acquires the title Sir or Dame?
Do they have any specific obligations towards the state? Or are they just basically people with high status and too much money and manpower, who can choose to cooperate with the state at their own leisure?

Prince (Knyaz, Князь) had a higher ranking. Most (well, excluding HUGE amount of georgian princes) of princes were Ryurik (russian) or Gedimin (lithuanian) descendants. Volkonskys are Ryurikovichi, Kurakins are Gediminovichi.
Direct familial relationship would be Great Prince. Though Ryurik descendants are from originally ruling family (Ryurikovichi).
Yes for both. Alexandr Surorov became a count Rymniksky (for victories aganst ottomans in 1787-1791) and latter a prince Italiysky (for victories against french in 1799). Though it was extremly rare to became a prince. Bezborodko wasn't from aristocraty, he became count during Catherine II ruling.
Originally they had obligation of state service, but it was cancelled by Peter III in 1762.

[Edited] PS. Marriage to count Bezukhov was advantageous because his dead father was extremely rich. Being noble didn't mean being rich.