Irish Grammer question T.G

Hi I'm doing my leaving cert this year and trying to get to grips with The tuiseal ginideach. e.g Bas an Tíogair Cheiltigh, comes from an Tiogar ceilteach. Are both tiogair and cheiltigh tuiseal ginideach and if not why does tiogar take an i?

Also for i rith an gheimridh fhuair. is fuair an adjective?? I was thought that adjectives only take a h if following a feminine word and is geimridh taking a h because it doesn't take a h when it is just an geimhreadh. Thanks.