I suspect this has already been discussed but I couldn't find it so if is I apologise for bringing it up again
Whenevr I think I understand grapple a PC does stuff that needs further thought.
If a PC grapples someone (X) on round 1 and X fails to escape on his turn:
- does the PC have to do anything on his next turn (give up an attack, make another roll?) to maintain the grapple?
- can the PC say hold X with his off hand and cast a spell, or strike him with his weapon?
- if X has multiple attacks does he get multiple chances to break the grapple?
- if X is prone is he at disadvantage to break the grapple?
- if X decides to "grab the PC's weapon arm" how is this resolved? (Seems like a grapple?)
- if the PC has a shield in his off hand can he use it to bash X with an improvised weapon? If so does he get to keep the shield bonus to AC?
ah grapple thou art ever an enigma.
Grappling, as defined in the rules, is fairly limited in scope and utility. It allows the Grappler to control the movement (and nothing else) of the Grappled creature. That's it. There is no reference to what exactly is grappled and what the Grappler needs to do to preserve the grapple. (Such as giving up a hand to hold the target, for instance.)
That being said, in every campaign I've DM'ed or played in, grappling has been ruled as a precondition for all sorts of different actions against other characters, including tying up arms, weapons, etc.
To answer your specific questions, as best I understand them:
"Does the PC have to do anything on his next turn..."
No. The grapple continues until the grappler chooses to end it, is incapacitated, moved out of range by a spell or effect (such as thunderwave or teleport), or the target successfully escapes the grapple as an action on his or her turn. (See Grappled condition on Basic Rules page 105.)
"Can the PC ... cast a spell or strike with his weapon?"
Yes. Because, as written, the grapple does not require the grappler to give up a free hand, they can continue to act normally once they are grappling somebody. Of course, the target can also act normally and attack whoever is grappling them without penalty, so long as they are able to make melee attacks.
"If X has multiple attacks, does he get multiple attempts to break the grapple?"
No. See Basic Rules pg 74: "A grappled creature can use its action to escape." It does not say that it uses its Attack action to escape. The text on Grappling explicitly allows the Grapple to substitute one of multiple attacks with the Attack action.
"If X is prone is he at disadvantage to break the grapple?"
No. The prone condition only give disadvantage to attack rolls, not ability checks. The grapple does, however, keep the target from standing. (Again, see Prone and Grappled conditions on Basic Rules page 105.)
"If X decides to grab the PC's weapon on arm, how is this resolved?"
This is a DM ruling. Personally, I would treat this as a specific grapple. If successful, the target would not be able to attack until the grapple is broken, and the grappler would need to use one or both of his or her hands to hold the weapon arm. (Depending on size of weapon.) A subsequent athletics check could possibly be made to try to disarm the PC, though I would probably impose disadvantage to the grappler. Grappling a sword would also probably be harder than grappling a quarter-staff. All up to the DM as the situation warrants, of course.
"If the PC has a shield in his off-hand, can he use it to bash X with an improvised weapon?"
Yes. As discussed elsewhere, Crawford's ruling on this seems to be that the Shield maintains its AC bonus when being used as an improvised weapon. On the other hand, because it does not resemble another weapon, the attack would probably be made without proficiency, unless the PC had the Tavern Brawler feat.
Again, it's pretty limited as covered in the rules. On the other hand, there's nothing stopping the DM from using the Grappling framework (opposed athletics check, option to use an Action to escape) as a way to cover any range of creative actions players might want their characters to attempt. Common sense should prevail. But if the player wants to accomplish more than just controlling the movement of their target, they should probably tie up at least one (if not 2) hands, and/or make multiple successful checks.
