Your friend was apprehended in 2018 for trying to apply for a visa at the USA embassy with a different identity from one she had used to apply to the same embassy in 2010. After being apprehended, she was fingerprinted by the USA embassy, convicted in court and jailed in Cote d'Ivoire for illegal acquisition of citizenship. Now she wants to apply for a UK entry clearance.
Will this captured fingerprint create a problem or not?
Yes it should. USA and UK share some immigration information via Five Eyes and other Treaties and have even taken steps in the last couple of years to share more.
And do they run fingerprints match against others taken elsewhere in
Europe or America?
They do, to what degree nobody knows however it will be foolhardy to test the system.
Finally she has a criminal record which should be disclosed on all visa applications when asked although typically embassies do not verify criminal records before issuing visas.
If she applies and tells the truth, she is not going to get a visa. If she lies, it is unlikely the conviction in Cote d'Ivoire will be discovered by the UK embassy (unless she triggers a background check) however because hers was a criminal case it is more likely than average that she is in the USA database which is shared with UK