RAW, anything a player character can do
The caster decides, what action the animated object will take:
You decide what action the creature will take and where it will move during its next turn, or you can issue a general command
This action does not have to be an attack. According to the Monster Manual (page 10 "Actions"), besides actions from the stats block, there are actions available to all creatures:
When a monster takes its action, it can choose from the options in the Actions section of its stat block or use one of the actions available to all creatures, such as the Dash or Hide action, as described in the Player's Handbook.
Therefore, animated objects can at least Dash, Disengage, Dodge, Help, Hide, Grapple and Shove.
They probably can do more, since the PHB describes Grapple and Shove as "the most common" (but not the only possible) contests:
This section includes the most common contests that require an action in combat: grappling and shoving a creature. The DM can use these contests as models for improvising others.
This might be table-dependent
Can silk ropes grapple and drag away enemies?
It can try, if the caster gives the respective order. As a player, you declare "I command my rope to drag away this guy" and wait for the DM to narrate the results. That's how the game works according to the "How to play" chapter of the PHB, page 6.
D&D 5th edition empowers the DM in ways that 3rd, 3.5, and 4th did not. While rule zero has always applied, 5th edition chooses not to explicitly codify many things.
Your DM might assume, a particular object (a ball, for instance) cannot grapple because it can't do this physically (it has no appendages, etc). If your DM says it can't, it can't. The DMG admits there are no rules for all the possible situations and suggests to resolve them using the common sense:
The rules don’t account for every possible situation that might arise during a typical D&D session. For example, a player might want his or her character to hurl a brazier full of hot coals into a monster’s face. How you determine the outcome of this action is up to you