Wolfram Language (Mathematica), 267 246 190 172 170 bytes

R[ToString[""@ToExpression@R[#,{"("->"~In~##&[",")"->"]",a:(WordCharacter|"_")..:>q<>a<>q}]/.#2/.$...[[0]]|List->(##&)/.In->(#@##2&),CForm],q->""]&
R=StringReplace
q="\""

Try it online!

Input ["expr", packs]. packs can be in any format accepted by ReplaceAll.

There unfortunately seems to be no straightforward way to simultaneously expand heads (Through, Comap^14.0+) and arguments (Thread).

R=StringReplace;q="\""
ToExpression@R[#,{                  parse input:
 "("->"~In~##&[",")"->"]",            convert f(args...) to In[f,args...]
 a:(WordCharacter|"_")..:>q<>a<>q     convert symbols to strings
}]
""@ %                               wrap in ""
  /.#2                                insert packs
  /.$...[[0]]|List->(##&)             expand packs
  /.In->(#@##2&)                      convert In[f,args...] to f[args...]
ToString[ % ,                       stringify,
  CForm                               converting [] to ()
]
R[ % ,q->""]                        unquote names

Python 3, 567 bytes

lambda v,q:w(g(v,p(q),-1))
def c(s):
	a=[];b=0;l=""
	for x in s:
		if x!=","or b:l+=x;b+=~-")(".find(x)%3-1
		else:a+=[p(l)];l=""
	return a+[p(l)]
p=lambda s:(0,p(s[:-3]))if"..."==s[-3:]else(1,(2,s.split("(")[0]),c(s[s.find("(")+1:-1]))if")"==s[-1]else(2,s)
def g(v,q,i):
	if q[0]>1:return q[1]if q[1]not in v or i<0else v[q[1]][i]
	if q[0]:return(g(v,q[1],i),[g(v,k,i)for k in q[2]])
	a=[];i=0
	while 1:
		try:a+=[g(v,q[1],i)];i+=1
		except:return a
w=lambda k:",".join(q for q in[w(x)for x in k]if q)if type(k)==list else k if type(k)==str else k[0]+"("+w(k[1])+")"

Try it online!

Could use a lot of golfing but I'm pretty happy with this answer. Tested on all 10 example cases.

-9 bytes thanks to Neil