1. How far or deep can a board of directors get involved in judging or influencing the leadership style and the leadership activities of its CEO?

The answer of course is that "it depends." Let us accept firstly the premise that a board appoints its CEO to fulfil certain tasks and to achieve certain outcomes. If the board is rational and competent, it determines the leadership and management skills and attributes needed to satisfy the board's objectives and appoints the best person it can afford who possesses those attributes to the rol. The reason is because such an appointment lowers appointment-risk and increases the likelihood of successfully achieving that which was intended.

Of course, if the board doesn't behave rationaly or competently, then the appointment may be sub-optimal or fail.

When we talk about "leadership and management skills" we inevitably include leadership style and "activities" or management practices. These too vary with the context.

The leadership style required for a predominantly "left-brain" organisation is different to one which is predominantly right-brain. Similarly, what you do and how you do it, from a leadership perspective, will vary, for example depending on whether you are changing an organisational culture, process or other corporate attribute very quickly or slowly (i.e. "big bang" change versus "continuous improvement".) Management style for revolutionary change is quite different to a "steady as she goes" type of change agenda.

To each of these myriad of contexts is applied a "management style" that will work best in that context.

If the CEO doesn't posses in his/her skill-set an ability to apply the optimal "style" then a sub-optimal outcome will occur. Sometimes this sub-optimal/dysfunctional approach can be very painful for an organisation.

The board's role, among other things, is to monitor CEO behaviour and performance. If sub-optimal leadership strategies are being applied within the orgaisation, the board, through the Chairman, should discuss the matter with the CEO in order to understand the issues: why did the CEO use that technique; is the CEO aware of the ramifications of that chosen technique; is the CEO capable of utilising the optimal strategy, etc

If the CEO doesn't satisfy the Chairman/board that they will/can remediate the situation in an acceptable time-frame, then the board may consider this one of the triggers that starts a search for a new CEO.

Remember that a CEO is meant to be "an expert" in dealing with and satisfying the challenges facing the organisation and for which that CEO was appointed. Board "involvement" and "influence" should only be applied when warranted, and that is a very subjective decision.

Unless the board is mature and knowledgable, then they too may misjudge the situation and impose themselves unjustifiably on the CEO when he/she is quite capable of handling the issues.

2. To what extent HR matters ( usually the domain of the CEO) can be treated on board level? 

The importance of HR will vary depending on the corporation, what it does and how it does it. I don't know of any company however that doesn't require people - so the importance of HR, culture, industrial relations, and so on will vary depending on context.

When the management of HR or HR-related issues threatens to destabilise the company, force the company to take on unjustified risk or expense, or when the issue threatens to  tarnish the company's reputation and ability to sustainably transact its business; then the board must interfere to ensure suitable remedies, strategies or solutions are in place. That doesn't mean that the board has to do it, but rather that it is done. If internal staff can't do it, then it's the board's responsibility to ensure that competent outsiders are brought in to "fix it".

3. What are tasks of board committees?

Each board committee (e.g. Audit, Finance, etc) has its own "agenda", own timelines, and its own objectives. The tasks therefore vary depending on those variables.

4. In what way is CEO to be involved in choosing his own successor etc?

It will vary from context to context. If the CEO has been successful, respected and is, say retiring on good terms or has been head-hunted into a new prestigeous role, then I suspect he/she will be actively involved in their own replacement.

If on the other-hand the incumbent has failed in his/her role, why would a board seek his/her involvement in the replacement?