This information is currently not contained in the rules and is generally in the purview of the DMG (which was not released at the time of writing this answer, but this more recent answer addresses).
Until then, I would class most objects as having a DC of 5 or so (10 - 5 for a Dex of 0). Armored or particularly hard objects would have an AC equivalent to roughly their amount of armor, so somewhere between 10 and 18.
So, I couldn't sleep and decided to work on that. Here's the best I got until the moment - using only PHB.
I have done the math for 3 builds so far, all using Variant Human and fighting style Archery when multiclassed:
- Rogue 3, Crossbow Expert, Hand Crossbow +1;
- Rogue 2/Fighter 1, Sharpshooter, Heavy Crossbow +1;
- Rogue 2/Fighter 1, Crossbow Expert, Hand Crossbow +1;
Sneak Attack procs are assumed every round, as there are melee allies. I'm the first to say I was not very creative with the use of subclasses or magic items. As noted by András in the comments, Sneak Attack only procs once per turn, so the second Hand Crossbow attack does not deal the extra 2d6 damage, putting both builds roughly 2.5 average points below the Sharpshooter.
Assuming you you can hide to get unseen every turn
If Hide = Unseen = Attack with advantage is a thing (this depends on your stealth checks, but with expertise they should be very likely to work), the highest damaging build becomes Rogue 2/Fighter 1, Sharpshooter, Heavy Crossbow +1.
- Cunning action for hiding.
- Attack with advantage. Damage on hit = \$5.5 (1\textrm{d}10) + 3.5 (1\textrm{d}6) + 10 + 5 = 24\$. Damage on critical = \$11 + 7 + 10 + 5 = 33\$.
- The probability of hitting without a crit is given by \$1 - (1 - \textrm{Usual probability})^2\$, where Usual probability is calculated using the +4 Modifier. Against ACs 16, 17, and 18, that works out to \$\{0.64, 0.5775, 0.51 \}\$ respectively.
- The probability of a crit with advantage is \$1 - 0.95^2 = 0.0975\$.
- The final average damage is \$ \{18.57, 17.07, 15.45 \} \$, for ACs 16, 17 and 18, respectively.
For the Rogue 3, Crossbow Expert build, assuming we can attack with the same hand crossbow twice, we have nine possible situations. Each one has a \$P_i\$ associated with it and an average damage \$ d_i \$. The average total damage is given by \$ \sum_{i=1}^{9} P_i \cdot d_i \$. Let \$ P = \{ 0.55, 0.5, 0.45 \} \$ be the probability of hitting an individual hit and \$ P_c = 0.05\$ the probability of landing a critical hit.
- First attack hits, second misses. \$ P_1 = P \cdot (1 - P - P_c) \$; \$ d_1 = 3.5 + 7 + 5 = 15.5\$.
- First attack misses, second hits. \$ P_2 = (1 - P - P_c) \cdot P \$; \$d_2 = 15.5\$.
- First attack crits, second misses. \$ P_3 = P_c \cdot (1 - P - P_c) \$; \$d_3 = 7 + 14 + 5 = 26 \$.
- First attack misses, second crits. \$ P_4 = (1 - P - P_c) \cdot P_c \$; \$d_4 = 26 \$.
- First attack hits, second crits. \$ P_5 = P \cdot P_c \$; \$d_5 = 3.5 + 7 + 5 + 7 + 5 = 27.5 \$.
- First attack crits, second hits. \$ P_6 = P_c \cdot P \$; \$d_6 = 7 + 14 + 5 + 3.5 + 5 = 34.5 \$.
- Both attacks crit. \$P_7 = P_c \cdot P_c\$; \$d_7 = 7 + 14 + 5 + 7 + 5 = 38 \$.
- Both attacks hit. \$ P_8 = P \cdot P \$; \$ d_8 = 3.5 + 7 + 5 + 3.5 + 5 = 24\$.
- Both attacks miss. \$P_9 = (1 - P - P_c) \cdot (1 - P - P_c)\$; \$d_9 = 0\$.
The result of the summation, if I did everything right, gives us an average damage of \$ \{16.92, 15.79, 14.62 \} \$.
Similarly, for Rogue 2/Fighter 1, removing the 1d6 from Sneak Attack, we get \$\mathbf{d} = \{12, 12, 19, 19, 24, 27.5, 31, 20.5\} \$, but increasing the Probabilities to \$ \{ 0.65, 0.6, 0.55\}\$, we get an average damage of \$ \{ 15.66, 14.70, 13.73 \} \$.
Best Answer
Animate Objects a 5th level spell should allow you to do this for 1 minute at a time.
My gnomish Wizard loved to beat people up with flying gold coins
Tiny Servant a 3rd level spell from Xanathar's Guide to Everything is another way to animate a tiny object for up to 8 hours.
Awaken a 5th level spell costs 1000 gp and is instantaneous. It can awaken a bush(bring me a shrubbery!) or a tree allowing it to move on its own. It is only charmed by you for 30 days.
Create Homunculus a 6th level spell from Xanathar's Guide to Everything turns blood, clay, ash and mandrake root into a special construct called a homunculus found in the Monster Manual.
True Polymorph a 9th level spell is yet another way to animate an object. however it does lose its original form so while transformed it IS a dragon not a house or castle and must be CR 9 or less. If you make this permanent it is no longer under your control.
and as always Wish a 9th level spell can do this as well.
Shield Guardian Amulet or a Figurine of Wondrous Power maybe what your looking for if you want a longer term companion. The Shield Guardian Amulet is found only in the Monster Manual (under Shield Guardian) though so you'll need to ask the DM for it. The Figurine of Wondrous Power comes in many forms (the Bronze Griffon is really nice) and is found in the Dungeon Master's Guide and may be available for sale somewhere.
Manual of the Golems is another expensive time consuming yet permanent way to animate an inanimate golem. It uses anywhere from 30 days and 50,000 gp to 120 days and 100,000 gp and consumes a very rare magic item, provided you have a spell caster with 2 5th level spell slots available on a daily basis.