It is said that for students of knowledge or simply a layman who does not have much Islamic knowledge, it is better to stick to the madh-hab of one’s local area.

What would be the ruling in case a person is acquiring knowledge from some other source (internet) and clearly sees ‘aqeedah issues in the local scholars?

Along the same lines, can one refer to the websites that one trusts, for example, islamqa.info, for general issues, instead of asking people of knowledge in one’s area for every situation because it is easier to get an answer or because we "personally" think it is more reliable?

Also, can one always choose to follow the opinion of any madh-hab in every situation, given that we are convinced of the evidence provided by the scholars, not by our own interpretation of the Qur'an and the Sunnah, and without choosing specific local madh-hab?

The ideal situation is when you and I are able to look into the Qur'an, deduce laws directly from it, and distinguish between a weak and a sahih hadeeth. I pray that this day comes soon.

‘Aqeedah of the person is very important and it is "the most essential thing". In most cases, the fiqh rulings of a person are in line with his ‘aqeedah. So I agree with you that we have to be careful about the ‘aqeedah of the person from who we are taking our knowledge from.

The point is that, due to a lack of knowledge, we are to follow a madh-hab, and this is where we are supposed to start. But we do not follow any madh-hab, we have to follow the madh-habs of any of the ahle sunnah wal jammah (the 4 main madh-habs).

If one of these madh-habs is followed in your area then that will be a good place to start. As you progress with your studies, you will have to incorporate your newly acquired authentic knowledge into your practices. So, for example, if you learn in your madh-hab that raising the hands in the takbirs of salah is not necessary and you later learn that it was indeed the sunnah of the Prophet (ﷺ) to raise the hands, then you will have to follow the sunnah despite what your madh-hab says. [1]

You have to be very careful about where you get your knowledge and information from. If the scholars in your area are trustworthy then you will have to consult them first because they can easily respond to your questions in a way that will be easy for you to understand. If, on the other hand, they are not trustworthy, you can look elsewhere. Islamqa.info is a good website and it is recommended by many shuyookh but we do not just refer to it because we can easily access it because we might not understand certain fatawa given on this site.

As for your last question, yes, we can do that. What is required from us is to take the direction the daleel leads us to. [2]

References:

[1] Dr. B. Philips, The Evolution of Fiqh, p. 58

[2] Dr. B. Philips, The Evolution of Fiqh, p. 151