Lucas Vs. The Garage Door Remote

Javascript (ES6), 277 275 263 253 250 247 234 bytes

_=>(d=[0,0],l=s='',[...Array(60)].map(_=>(s+=`
${c=Math.random()*3|0}:(`,l+=c,d=d.map((v,i)=>(v=v&8?v&16?v-27?v+1:20:v-9?v-1:0:v,v^=c+~i?0:v&8||24,s+='LR'[i]+`:${(v&7)*25}% `+(v&8?v&16?'opening':'closing':'stopped')+',)'[i],v)))),l+s)

Ungolfed and commented

_ => (
  // Initialize array:
  //   - d = door states as integers
  //     - bits 0 to 2: door opening state (from 0b000 = 0% to 0b100 = 100%)
  //     - bit #3: door in motion (0: no, 1: yes)
  //     - bit #4: door direction (0: closing, 1: opening)
  d = [0, 0],

  // Initialize strings:
  //   - l = list of commands
  //   - s = door states in plain text
  l = s = '',

  // Iterate on an array of 60 entries.
  [...Array(60)].map(_ => (
    // c = new random command (0, 1 or 2)
    // Append new line and new command to s.
    s += `\n${c = Math.random() * 3 | 0}:(`,

    // Append new command to l.
    l += c,

    // For each door ...
    d = d.map((v, i) => (
      // If the door is in motion, update its opening state.
      // Clear the 'in motion' bit if a bound is reached (either closed or fully open).
      v = v & 8 ? v & 16 ? v - 27 ? v + 1 : 20 : v - 9 ? v - 1 : 0 : v,

      // If the current command is intended for this door, update its direction and
      // 'in motion' bit. Direction is changed on the 'stopped => moving' transition.
      v ^= c + ~i ? 0 : v & 8 || 24,

      // Translate the door state in plain text and append it to s
      s +=
        'LR'[i] +
        `:${(v & 7) * 25}% ` +
        (v & 8 ? v & 16 ? 'opening' : 'closing' : 'stopped') +
        ',)'[i],

      // Value to be taken into account by map()
      v
    ))
  )),

  // Final result to be returned
  l + s
)

Demo

let f = 
_=>(d=[0,0],l=s='',[...Array(60)].map(_=>(s+=`
${c=Math.random()*3|0}:(`,l+=c,d=d.map((v,i)=>(v=v&8?v&16?v-27?v+1:20:v-9?v-1:0:v,v^=c+~i?0:v&8||24,s+='LR'[i]+`:${(v&7)*25}% `+(v&8?v&16?'opening':'closing':'stopped')+',)'[i],v)))),l+s)

console.log(f())

Python 2, 377 370 361 357 345 335 326 316 312 306 304 bytes

The second indent level is a raw tab (\t), which plays really badly with Markdown, so it's been replaced by two spaces.

from random import*
p=[randint(0,2)for d in[[0]*3]*60]
print`p`[1::3]
v=[-1]*3
c=[0]*3
f=lambda y:str(c[y]*25)+'% '+'csoltpooespnipinengdg'[d[y]+1::3]
for x in p:
 for i in 1,2:
  q=d[i];c[i]+=q
  if(2*-~q==c[i])*q:v[i]=q;d[i]=0
 z=d[x]
 if z:v[x]=z
 d[x]=-v[x]*(z==0);print'%d: (L:%s, R:%s)'%(x,f(1),f(2))

I'm almost certain this can be golfed further.

Ungolfed, with comments:

import random

# Generate the random string - represented as a list of ints
presses = [random.randint(0, 2) for _ in range(60)]
print ''.join(map(str, presses))

# Constants for door states used for easier reading
CLOSING = -1
STOPPED = 0
OPENING = 1

# Variables representing the state of the garage doors
# There's a third element in these so that x[0] resolves to a dummy slot
# (this way, we can avoid a conditional down the road)
prev_states = [CLOSING, CLOSING, 0]
door_states = [STOPPED, STOPPED, 0]
door_pcts = [0, 0, 0]  # delta 1 = 25%

for press in presses:
  # Close/open the door 1 more
  for i in 1, 2:
    if door_states[i] != STOPPED:
      delta_pct, stop_pct = (-1, 0) if door_states[i] == CLOSING else (1, 4)
      door_pcts[i] += delta_pct
      if door_pcts[i] == stop_pct:
        prev_states[i] = door_states[i]
        door_states[i] = STOPPED

  # Handle pressing a button
  # If the press is 0 (no press), the 0th element resolves to a dummy
  # door, thus saving us an expensive conditional

  if door_states[press] == STOPPED:
    door_states[press] = -prev_states[press]
  else:
    prev_states[press] = door_states[press]
    door_states[press] = STOPPED

  # Print the status update
  print '%d: (L:%d%% %s, R:%d%% %s)' % (
    press,
    door_pcts[0]*25,
    ['closing', 'stopped', 'opening'][door_states[0]+1],
    door_pcts[1]*25,
    ['closing', 'stopped', 'opening'][door_states[1]+1],
  )

Saved 4 14 15 bytes thanks to @TheBikingViking!

Saved 6 bytes thanks to @ValueInk!

C, 420 433 424 374 bytes

#define F(X,x) X=x==1?X+1:x==2?X-1:X;X=X<0?0:X>4?4:X;x=X==0?0:X==4?3:x
#define G(x) x=x==1?3:x==2?0:x==3?2:x+1
#define H(X,x) X*25,x==0||x==3?"stopped":x==1?"opening":"closing"
c,i,l,r,L,R,x[60];main(){while(i<60)printf("%d",x[i++]=random()%3);while(c<60){if(x[c]==1)G(l);else if(x[c]==2)G(r);printf("\n%d: (L:%d%% %s, R:%d%% %s)",x[c++],H(L,l),H(R,r));F(L,l);F(R,r);}}

Doesn't seed the random generator but does use random for a better distribution. There must be a better way to golf this logic though...

110121100121212100112200222111200020111100022122202112202211002
1: (L:0% opening, R:0% stopped)
1: (L:25% stopped, R:0% stopped)
0: (L:25% stopped, R:0% stopped)
1: (L:25% closing, R:0% stopped)
2: (L:0% stopped, R:0% opening)
1: (L:0% opening, R:25% opening)
1: (L:25% stopped, R:50% opening)
0: (L:25% stopped, R:75% opening)
0: (L:25% stopped, R:100% stopped)
1: (L:25% closing, R:100% stopped)
2: (L:0% stopped, R:100% closing)
1: (L:0% opening, R:75% closing)
2: (L:25% opening, R:50% stopped)
1: (L:50% stopped, R:50% stopped)
2: (L:50% stopped, R:50% opening)
1: (L:50% closing, R:75% opening)
0: (L:25% closing, R:100% stopped)
0: (L:0% stopped, R:100% stopped)
1: (L:0% opening, R:100% stopped)
1: (L:25% stopped, R:100% stopped)
2: (L:25% stopped, R:100% closing)
2: (L:25% stopped, R:75% stopped)
0: (L:25% stopped, R:75% stopped)
0: (L:25% stopped, R:75% stopped)
2: (L:25% stopped, R:75% opening)
2: (L:25% stopped, R:100% closing)
2: (L:25% stopped, R:75% stopped)
1: (L:25% closing, R:75% stopped)
1: (L:0% opening, R:75% stopped)
1: (L:25% stopped, R:75% stopped)
2: (L:25% stopped, R:75% opening)
0: (L:25% stopped, R:100% stopped)
0: (L:25% stopped, R:100% stopped)
0: (L:25% stopped, R:100% stopped)
2: (L:25% stopped, R:100% closing)
0: (L:25% stopped, R:75% closing)
1: (L:25% closing, R:50% closing)
1: (L:0% opening, R:25% closing)
1: (L:25% stopped, R:0% stopped)
1: (L:25% closing, R:0% stopped)
0: (L:0% stopped, R:0% stopped)
0: (L:0% stopped, R:0% stopped)
0: (L:0% stopped, R:0% stopped)
2: (L:0% stopped, R:0% opening)
2: (L:0% stopped, R:25% stopped)
1: (L:0% opening, R:25% stopped)
2: (L:25% opening, R:25% closing)
2: (L:50% opening, R:0% opening)
2: (L:75% opening, R:25% stopped)
0: (L:100% stopped, R:25% stopped)
2: (L:100% stopped, R:25% closing)
1: (L:100% closing, R:0% stopped)
1: (L:75% stopped, R:0% stopped)
2: (L:75% stopped, R:0% opening)
2: (L:75% stopped, R:25% stopped)
0: (L:75% stopped, R:25% stopped)
2: (L:75% stopped, R:25% closing)
2: (L:75% stopped, R:0% opening)
1: (L:75% opening, R:25% opening)
1: (L:100% closing, R:50% opening)

Older version 1:

c,i,l,r,L,R,x[60];main(){while(i<60)printf("%d",x[i++]=random()%3);while(c<60){if(x[c]==1)l=l==1?3:l==2?0:l==3?2:l+1;else if(x[c]==2)r=r==1?3:r==2?0:r==3?2:r+1;printf("\n%d: (L:%d%% %s, R:%d%% %s)",x[c++],L*25,l==0||l==3?"stopped":l==1?"opening":"closing",R*25,r==0||r==3?"stopped":r==1?"opening":"closing");L=l==1?L+1:l==2?L-1:L;R=r==1?R+1:r==2?R-1:R;L=L<0?0:L>4?4:L;R=R<0?0:R>4?4:R;l=L==0?0:L==4?3:l;r=R==0?0:R==4?3:r;}}

Older version 2:

c,i,l,r,L,R,x[60];g(){while(i<60)printf("%d",x[i++]=random()%3);}main(){g();while(c<60){if(x[c]==1)l=l==1?3:l==2?0:l==3?2:l+1;else if(x[c]==2)r=r==1?3:r==2?0:r==3?2:r+1;printf("%d: (L:%d%% %s, R:%d%% %s)\n",x[c++],L*25,l==0||l==3?"stopped":l==1?"opening":"closing",R*25,r==0||r==3?"stopped":r==1?"opening":"closing");L=l==1?L+1:l==2?L-1:L;R=r==1?R+1:r==2?R-1:R;L=L<0?0:L>4?4:L;R=R<0?0:R>4?4:R;l=L==0?0:L==4?3:l;r=R==0?0:R==4?3:r;}}