How strong would robin have to be to be able to pull off all of those tricks with the Hana Hana no Mi since she's using her own arm? I mean like she can create webs than can stop canon balls, break soldier's backs and use hands to create huge arms/legs and be able to control them. How strong are Robin's muscles?
Let us investigate this. Nico Robin is human, who ate a paramecia type devil fruit. She should be as strong as a normal human (physically) Wiki corroborates it so far.
Robin has no superhuman abilities beyond her Devil Fruit powers but is an above average athlete. While the frequent use of her powers in battle leave much of her physical attributes in question, it is hinted from time to time that Robin's physical prowess is actually much stronger than her build would otherwise suggest.
Robin's greater than average feats include (copying directly from the wiki so I don't miss any)
She was seen being able to take on direct hits from both Yama's drop kicks and punches (with the latter sending her flying through a wall, destroying it) and was still able to effectively retaliate after maneuvering the latter away from the ruins of Skypiea.
She was able to restrain a powerful zombie like Jigoro (albeit briefly). She was also able to survive being impaled through the chest by Crocodile and remain alive without medical attention from a doctor, until the wound healed on its own.
She is agile enough to maneuver (while maintaining her arms' "X" formation) her way through a group of Marine captains and commanders before using her powers in retaliation. She has great recovery ability, for example when she, Zoro, Gan Fall, and Wiper were all knocked out by Enel's Devil Fruit ability, Robin was the first person to wake up. Even with a severe injury, she still had enough strength to use her Devil Fruit ability to carry them to safety and had fully recovered without the aid of Chopper.
One of her most impressive aspects is her will power; as she managed to recover in mere minutes after having her shadow stolen by Gekko Moriah (when it usually takes a few days for most humans).
Thus Robin has above average agility, strength, will power and recovery.
Moving on to the powers of her DF; Hana Hana no Mi
The Hana Hana no Mi is a Paramecia-type Devil Fruit that allows the user to replicate and sprout pieces of their body from the surface of any object or living thing
We've seen Robin is very intelligent, probably the most, in the Straw Hat Crew. She uses her Df in a very effective manner. She doesn't rely on strength but on blocking the movement of the enemies and intelligently using holds and distractions.
As she claims to Pell, speed and strength mean nothing to her, as she can render almost any foe into submission with ease, even those with superhuman speed such as Hakuba
However, her main weakness is,
The user will gain damage to their body if the extra limbs are attacked, and can feel the pain on each one of them. This disadvantage also extends to other Devil Fruits, as if her limbs are affected, then the host body is affected as well, as demonstrated when Sugar touched her limbs with her Hobi Hobi powers, turning her real body into a toy.
The strength of the user is also still limited to their individual strength, meaning that the individual strength in the replicating appendages is no different to that of Robin's actual corresponding limbs.
The last line is of the main importance. EACH of the appendage can exert strength equal to that of main body. Thus Robin utilizes strength of multiple arms to produce the required result.
Breaking Soldier's backs; Clutch: Robin doesn't use strength as much as her knowledge. These holds turn the trapped soldiers'strength against them. These type of submission holds are common in MMA, wrestling etc. And can easily dislocate joints or break bones.
Stopping Cannonballs: This is the most contentious. It can be argued that multiple arms work together to stop the cannonballs, but her real body should feel the pain of being hit by a cannonball. But being above average in strength and endurance, Robin can manage. I'll chalk it up to the edge conditions.
Giant Arms and Legs: Robin feels the fatigue from these giant creations. Combining these many arms can have the desired strength but flying shouldn't be possible. Robin explains its more of floating though. But her training in the timeskip allows her to use this technique for longer periods of time due to her endurance and willpower but not the strength.
TL;DR: From what we've seen Robin has above average physical prowess, but its her knowledge, effective technique and use of DF, and superhuman Will power, recovery and endurance that allow her to do the feats shown in One Piece.