Learn Polish: Czy dziedziczysz takie same cechy jak twój ojciec? - Do you inherit exactly the same traits as your father?

Taki sam (the same) has to match the noun it describes in gender, number, and case. Here, it describes cechy (traits), which is a non-masculine-personal plural noun in the accusative case. Therefore, we need the matching plural form, takie same.

Cechy is in the accusative case, so why is twój ojciec in the nominative?

In comparisons using jak, the noun after jak takes the case of its role in the implied second half of the sentence. The full thought is 'Do you inherit the same traits as your father [inherits]?' Since your father is the subject doing the inheriting in that implied second half, twój ojciec stays in the nominative case.

Could I just say takie cechy jak without same?

Yes, but it changes the meaning slightly. Takie same cechy jak means 'the exact same traits as'. If you drop same and just say takie cechy jak, it means 'such traits as' or 'traits like'.

What is the dictionary form of the verb dziedziczysz?

The infinitive is dziedziczyć (to inherit). It is an imperfective verb. It belongs to a family of words related to inheritance and descendants, sharing a historical root with words like dziedzic (heir) and dziecko (child).