They are both fine but have completely different meanings. On Sunday morning(s), I run in the park. (This describes something you do in the past, present and plan to do in the future. The ‘s’ is optional. No ‘s’ is the most general and categorical, suggesting you almost always do it. With an ‘s’ it is still a general habit but, imprecisely, perhaps not quite as consistent.) On Sunday morning, I ran in the park. (Once in the past. The default meaning would be last Sunday. However, if the context were that you were talking about a specific week in the past, it would mean Sunday of that week.)

Yes, both are OK. "On" is unnecessary, especially with the present tense: (On) Sunday mornings I run in the park. (habitual) (On) Sunday morning I run in the park. (habitual or future) (On) Sunday morning I ran in the park. (past) (On) Sunday mornings I ran in the park. (habitual past) Precise meaning may depend on context. If someone says to you "Sunday morning I run in the park", they might be talking about next Sunday (future). Or, they might be saying that running in the park is something they normally do on Sundays. The only way to know which meaning is intended is to listen to the other things they say.

Both "I run in the park" and "I ran in the park" can work in your sentence, but they have different meanings because they use different tenses: Simple Present Tense ("I run"): "On Sunday morning, I run in the park" implies that this is a regular activity or habit. This form suggests you do this every Sunday morning. Simple Past Tense ("I ran"): "On Sunday morning, I ran in the park" refers to a specific Sunday in the past. This tense is used to describe a completed action, meaning you ran in the park this past Sunday or on another Sunday in the past. In general, simple present is for habitual actions or routines, while simple past describes something that happened at a specific time in the past.