I'm having painless mouth sore - worried that it is syphilis or herpes infection, or oral cancer?

No worries yet: How long has the ulcer been present? A herpetic infection looks like a blister that has popped. The ulcer heals within about 10 days. Oral syphilis usually occurs on the lip or tongue and will heal on its own within about 3 weeks. The syphilis needs to be treated to prevent secondary or tertiary symptoms. Oral cancer can also look like an ulcer, but this will not heal.

Simply: Have your own general dentist or an oral surgeon take a look at what you are describing so that you will know exactly what it is and whether or not it needs treatment. From what you describe it's unlikely it's any of the three things you mentioned. However, you should find out what it is and the only way to do that is to have someone personally examine you.