Hey Yehoshua,
this stackoverflow post has great explanation of your case.
https://stackoverflow.com/questions/4377525/using-more-than-one-index-per-table-is-dangerous
I have table that has around 20 attributes (around 400,000 objects) and almost every attributes is indexed I didn't have any issue on updating that table only issue was with deleting objects from table and that wasn't related to index it was because of cascaded delete option.
Best regards, Slavko
As far as I have an understanding of how indexes work (my experience is in another developing language so other type of database but guess the bases will be the same) .... are 2 indexes enough in your case.
So 1 on Name and 1 on City (but only usefull when you have a lot of data in the entity and the search is performed frequently and needs to be quick)
I have no idea if your type of database automatically combines different indexes on the same entity (so that would make it even more performant) but again 2 indexes should be enough