Correct spelling of names, Chebyshev and Cholesky

Here you can hear the pronounce by a Russian speaking person.

As to the romanization, which usually does have a standard form in any language, I'd use the one of the language you are writing in your paper. In English it is Chebyshev (not Cheby_ch_ev). You can see the corresponding versions switching to other languages from the linked article in en.wikipedia (list of languages on the left).

According to ISO 9 (see also wikipedia), Чебышëв must be spelled Čebyšëv. Sometimes this spelling is used, cf. google.

Why not use the Cyrillic script? That is the canonical way of writing I would guess. (Leaving aside that Cholesky was French; his parents moved from Poland to France, according to wiki.)

All transcriptions are non-canonical, but equivalent up to (canonical?) isomorphism.

To quote wiki on Чебышёв:

Pafnuty Lvovich Chebyshev (Russian: Пафну́тий Льво́вич Чебышёв, IPA: [pɐf'n̪utʲɪ(j) 'lʲvovʲɪt͡ɕ t͡ɕɪbɨ'ʂof]) (May 16 [O.S. May 4] 1821 – December 8 [O.S. November 26] 1894)[1] was a Russian mathematician. His name can be alternatively transliterated as Chebychev, Chebysheff, Chebyshov, Tchebychev or Tchebycheff, or Tschebyschev or Tschebyscheff (the latter two pairs are French and German transcriptions).

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Finally, using non-roman script in LaTeX is not so hard anymore, these days.