Can a King of the House of David (i.e an inherited - not appointed - King), serve as king while his father is still alive?
I think King Shlomo might have been king during his father's lifetime.
Exactly. It is explicit in the beginning of Melachim Aleph (1:30) that King David ordered for Solomon to reign in his lifetime, which answers your question.
כִּ֡י כַּאֲשֶׁר֩ נִשְׁבַּ֨עְתִּי לָ֜ךְ בַּיהוָ֨ה אֱלֹהֵ֤י יִשְׂרָאֵל֙ לֵאמֹ֔ר כִּֽי־שְׁלֹמֹ֤ה בְנֵךְ֙ יִמְלֹ֣ךְ אַחֲרַ֔י וְה֛וּא יֵשֵׁ֥ב עַל־כִּסְאִ֖י תַּחְתָּ֑י כִּ֛י כֵּ֥ן אֶעֱשֶׂ֖ה הַיּ֥וֹם הַזֶּֽה׃
As I swore unto you by the Lord, the God of Israel, saying: Assuredly Solomon your son shall reign after me, and he shall sit upon my throne in my stead; verily so will I do this day.’