Yes, it's both correct and common to start a sentence with the 'ing' form of a verb, including the verb 'have'.
The verb 'have' does many jobs in English. It can indicate 'ownership':
e.g.' Having two jobs can be tiring. '
It can have many other meanings, often replacing other verbs, such as 'drink':
e.g. 'Having a cup of coffee after a meal can help you digest.'
It can be an auxiliary verb:
'Having seen the weather forecast, we decided to postpone our trip to the beach.'
And yes, you can start a sentence with 'Having got', but it doesn't mean the same as 'Having'.
If you say 'Having got', we would interpret 'Having' as a present participle of an auxiliary verb, and 'got' as the past participle of the verb 'get', meaning arrive, become or obtain, or as part of a phrasal verb. For example:
'Having got up earlier than usual, he decided to walk to work.'
This means 'Because he had got up earlier...'
Remember that 'have got' is only an alternative to 'have' in the present simple tense. 'Got' can't be used in other tenses of 'have' or with other forms of 'have'. The form 'Having' is either a gerund or a present participle, depending on the context, and you can't use 'got' after 'having' to indicate ownership.