I will answer to each element of your quesiton separatly:
Calculus (Tarter) is calcified bio-film and plaque that is stuck on the surfaces of your teeth. Inside are bacterias, dead or alive, along with their byproducts (and their toxins) that irritate the gums, which may eventually cause an immune response in the gums that will lead to a recession.
Therefore, scaling (removing) the "half" that is above the gums will do little or nothing to improve your situation, especially if there is anything left behind, bellow the gums.
The same can be said of any mouthrinces and other products you might try. They will hardly have any effect on the surfaces of the teeth above the gum line, due to the relatively short amount of time most people keep mouthrince in their mouth, the slow diffusion of the active ingredients across the calculus. It will not even reach below the gums wont remove/dissolve sub-gingival plaque or calculus.
On the other hand, if you chose to attempt to perform scaling deeper with picks, you risk damaging your gums, especially since the "picks" will be pointed towards your gums, and you will eventually slide them too far and therefore damage even more the soft tissues that surround and support your tooth, potentially accelerating the recession, or worsening the inflamation.
Dentists and hygienists use scalers and curettes, which cause minimal damage to the gums if used properly. Do not attempt to use them at home, on yourself, if you don't know what you are doing or lack good dexterity and technique. There are numerous videos on YouTube about how to use them.
Needless to say, if you have access to a hygienist, a dentist or periodontist (dentist that specializes in the treatment of gums) you will be in good hands.