1. You may write "ran quickly" --it makes total sense. The verb "sprint" does mean run quickly, but only for a short distance, but many times you do hear them used interchangeably. If you were writing a story and you used ran quickly in one sentence and sprinted in another then that would make the story more interesting because you're using different verbs that are basically synonyms.
2. When you tell a story, it's usually about the past. Use past tense.
For example past tense: "Yesterday after school, I ran quickly to my mom's house." (Sprinted would work in place of ran quickly, but I personally wouldn't use it.)
For example present participle: "I am running quickly to my mom's house." This means at this EXACT moment or in the next few moments you are or will be running.
For example past participle "I was running to my mom's house when I saw the dog." "I was running to my mom's house." IS WRONG. You need some more details after the past participle. I am a native English speaker from the USA.