Learn Turkish: Gerçi yazım kötü ama hikâye tutarlı. - Mind you, the spelling is bad, but the story is consistent.

Yes, they form a very natural pair. Gerçi concedes a minor negative point (meaning 'mind you' or 'granted'), while ama introduces the main, contrasting positive point. Together they create the common structure 'I admit X is true, but Y makes up for it'.

Where else could gerçi be placed in this sentence?

As a discourse marker, gerçi is highly flexible. While starting the sentence with it is standard, you could also place it right after the conceded point (Yazım kötü gerçi, ama hikâye tutarlı) to make it sound a bit more conversational.

Could yazım also mean 'my writing' or 'my summer'? How do we know it means 'the spelling' here?

Grammatically, yazım could indeed mean 'my summer' (yaz + -ım) or 'my writing' (yazı + -m). However, Turkish has a dedicated dictionary noun yazım which specifically means 'spelling' or 'orthography'. The context of critiquing a written story makes 'spelling' the obvious choice.

What is the root of the word tutarlı?

It traces back to the verb tutmak, which means 'to hold' or 'to catch'. This root extends to the concept of elements holding together properly, which gives us the adjective tutarlı, meaning 'consistent', 'coherent', or 'logical'.