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The rules are that one with Niederlassungserlaubnis (aka settlement permit) needs permission to stay away longer but unlike an unlimited residency permit, it is not automatically seen as abandoned. One needs to show the reasons you are leaving and a clear intent to return AND being able to do so without needing public assistance. This will likely mean having a job waiting for one upon return. Thus taking work elsewhere for some time is not unusual - but having lost a job and maybe not having one upon return is problematic.

Of course it is important that you have paid your taxes and social obligations but this doesn’t guarantee a right to return – on the contrary a lack of doing so would likely mean little tolerance for an extension. You don’t necessarily have to inform anyone about working abroad but as long as you are registered in Germany then you would be expected to pay tax on any income no matter where it is earned. So it is in your interest to make it clear you will be residing in another country for a while to avoid double taxation. But how long they might allow one to be away without losing the settlement will vary and need to be clarified with officials. I think if one is in another Schengen country, then one can be allowed up to 5 years. If outside of Schengen or the EU it will be shorter.

Alfie Jam wrote:

1. How can I request ABH to grant me an extension for staying outside EU for upto a year or preferably more?

My understanding is that an extension of stay abroad (from the six months allowed by default) is only given if the stay is of clearly defined temporary nature - e.g. your German employer send you abroad for three years. Your post sounds like this is uncertain in your case - which means you will lose your Niederlassungserlaubnis after six months abroad.

Alfie Jam wrote:

2. Why is physical presence in the country given so much importance if a person is registered, paying taxes & health insurance?

You can only be registered if you are physically present. If you keep your registration although you are absent (for more than two weeks, by law), you can be fined.
German health insurance will also not cover you abroad (at least outside the EU), so there is no point keeping it.
And, last not least, if you pay taxes although you don't have to (only residents are taxed), it is a voluntary gift to the German society (Thanks for that!), but will not give you any brownie points with the authorities - especially if it is based on an illegal act like wrongful registration!

Alfie Jam wrote:

5. Am I legally obligated to inform ABH about working abroad?

Yes - and again not doing so is a punishable offense.