Umlauts signify a specific vowel pronunciation as I recall, so are in all parts of names and words. Eg, das mädchen = the girl (which can also be used as a forename, eg Mädchen Amick) for German. That said Jääskeläinen is a Finnish name I think, so its not an umlaut if I understand it correctly, but a Scandinavian character used in the Finnish alphabet (if we still have some Finnish speaking posters they may be able to elaborate further).
Diacritical marks (like the umlaut or acute accent) are typically used to represent something that differs from the non marked usage. The acute accent in Spanish from my memory is used to indicate an irregular vowel stress or a distinction between words otherwise spelled the same. So González having an acute mark in her name in the text & picture should indicate pronunciation as gon-ZAH-les as opposed to Gonzalez which would be pronounced gon-zah-LES as I recall (however as we all know names aren't always pronounced how they're spelt, so...). A Spanish first name with an acute accent is Adrián, used to indicate the ia doesn't diphthong IIRC.