1. If a pianist were to perform a copyrighted work, especially contemporary piano music, in a public recital or concert, would the pianist or the concert's producer need to pay the copyright holder a fee to secure the rights to perform the copyrighted material in a recital? Does it make a difference whether the concert/recital is free or one that charges an admission fee?
I am answering from a USA perspective -- other countries may have different laws.
Fair Use copyright laws state that if someone were to sue you for the infringement of copyrights, they evaluate the use based on four points:
(1) the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
(2) the nature of the copyrighted work;
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
(4) the effect of the use upon the potential market for or value of the copyrighted work.
The actual use would be you reading the music. Your performance itself is basically a new work of art. Even if that wasn't true, #4 on the list could easily be argued to state that the music sales would go up based on your performance.
IOW, I don't think you should worry about it.
2. Related to 1, can the performance be recorded (video and audio) ? Would separate copyright arrangements be needed to facilitate recording of the recital/concert with copyrighted contemporary music?
Your performance is your own work of art. Just because it is based off someone else's work, you aren't required to have prior permission to record/publish your work. It is required that you credit the composition to the composer, and it is generally considered good practice to alert the composer or even ask permission to publish. However, those two things are not required -- they just lead to less possibility that the composer would try to seek out a lawsuit.
3. Suppose the pianist or the concert/recital's producer wants to publish the recording of the concert/recital for sale, or want to post some of the recording on the Internet, would additional copyright arrangements be needed then?
Well, the publisher of your work would gain a copyright for that performance simply by recording it. You aren't required to give any royalties back to the composer for performing their piece.
4. How would one go about securing the rights to perform, to record, and to distribute/sell the recordings?
Just do it

Any applicable knowledge, experience, and insights you share in this regard will be much appreciated. Thanks for your help in advance.
Well, my disclaimer follows: IANAL (I am not a lawyer), so don't take my words as being the law itself. There are quite a few websites that explain fair use and copyright laws -- one of the best is this one:
https://www.copyright.gov/fls/fl102.html. It was one of my sources when writing my paper on how the DMCA contradicts fair use rights.
The most useful section is this:
Copyright protects the particular way an author has expressed himself; it does not extend to any ideas, systems, or factual information conveyed in the work.
The safest course is always to get permission from the copyright owner before using copyrighted material. The Copyright Office cannot give this permission.