The sentence is specifically about the gloves that Ana gave you, not Ana's gloves.
"Ana's gloves are small" = "Mănușile lui Ana sunt mici" / "Mănușile Anei sunt mici"
You should probably use the first version ("lui Ana"), because applying a genitive/possessive form to a Romanian name can be quite vexing for non-natives, and it doesn't work for all names anyway.
Depends on the people. If you check out some non-English language wikis on the British Royal Family you'll note there every language has at least some of their names translated. Also depends on the historic period. Romanian wiki's page on the British royals, for example, seems to translate anyone who is older then prince Charles but leaves people Charles age and younger untranslated.
So if you're reading history books it is very very useful to know that Romanian Ana is actually the same as English Anne.