JavaScript (ES8),  266  262 bytes

s=>` _2
 1
 -2
 7\\ 3^__^
 8\\  (oo)\\_7
 11(__)\\ 7)${q=`\\/\\
 15||<>`}-4w |
 15|| 5||`.replace(/.(>|\d+)/g,(s,n)=>n^1?s[i=0].repeat(n^2?n:w):a.map(s=>q[j='7019'[!i++*2|!a[i]]]+s.padEnd(w)+q[-~j]).join`
`,a=s.split`
`.map(w=s=>` ${w>(v=s.length+2)?0:w=v,s} `))

Try it online!

How?

We first split the input string into lines surrounded with 2 spaces and compute the width w of the biggest line (including said spaces):

a = s.split`\n`.map(w = s => ` ${w > (v = s.length + 2) ? 0 : w = v, s} `)

We define q as:

"\/\[LF] 15||<>"
 ^^        ^ ^
 01        7 9

This string is used simultaneously as a substring of our template (except the <> part which is removed) and as a lookup string for the surrounding characters of the text bubble, using the marked positions above.

Each number n in the template string is interpreted as follows:

• 2: repeat the preceding character w times

• 1: insert the text bubble:

  a.map(s =>
    q[j = '7019'[!i++ * 2 | !a[i]]] +
    s.padEnd(w) +
    q[-~j]
  )
  .join`\n`
  

• anything else: repeat the preceding character n times

We actually match /.(>|\d+)/g, which means that the substring <> is interpreted as '<'.repeat('>'), resulting in an empty string.

JavaScript (ES8),  266  252 bytes

s=>` _2
 3
 -2
 5\\ 1^__^
 10\\  (oo)\\_5
 9(__)\\ 5)\\/\\
 13||-6w |
 13|| 7||`.replace(/.(\d+)/g,(s,n)=>n^3?s[i=0].repeat(n^2||w):a.map(s=>'|\\/<'[j=!i++*2|!a[i]]+s.padEnd(w)+'|/\\>'[j]).join`
`,a=s.split`
`.map(w=s=>` ${w>(v=s.length+2)?0:w=v,s} `))

Try it online!

How?

We first split the input string into lines surrounded with 2 spaces and compute the width w of the biggest line (including said spaces):

a = s.split`\n`.map(w = s => ` ${w > (v = s.length + 2) ? 0 : w = v, s} `)

Each number n in the template string is interpreted as follows:

• 2: repeat the preceding character w times (for the top and bottom borders of the text bubble)

• 3: insert the text bubble:

  a.map(s =>
    '|\\/<'[j = !i++ * 2 | !a[i]] +
    s.padEnd(w) +
    '|/\\>'[j]
  )
  .join`\n`
  

• anything else: repeat the preceding character n XOR 2 times

JavaScript (ES8),  266  263 bytes

s=>` _2
 1
 -2
 7\\ 3^__^
 8\\  (oo)\\_7
 11(__)\\ 7)${q=`\\/\\
 15||<>`}-4w |
 15|| 5||`.replace(/.(>|\d+)/g,(s,n)=>n^1?s[i=0].repeat(n^2?n:w):a.map(s=>q[j=i++?a[i]?7:0:a[1]?1:9]+s.padEnd(w)+q[j+1]).join`
`,a=s.split`
`.map(w=s=>` ${w>(v=s.length+2)?0:w=v,s} `))

Try it online!

How?

We first split the input string into lines surrounded with 2 spaces and compute the width w of the biggest line (including said spaces):

a = s.split`\n`.map(w = s => ` ${w > (v = s.length + 2) ? 0 : w = v, s} `)

We define q as:

"\/\[LF] 15||<>"
 ^^        ^ ^
 01        7 9

This string is used simultaneously as a substring of our template (except the <> part which is removed) and as a lookup string for the surrounding characters of the text bubble, using the marked positions above.

Each number n in the template string is interpreted as follows:

• 2: repeat the preceding character w times

• 1: insert the text bubble:

  a.map(s =>
    q[j = i++ ? a[i] ? 7 : 0 : a[1] ? 1 : 9] +
    s.padEnd(w) +
    q[j + 1]
  )
  .join`\n`
  

• anything else: repeat the preceding character n times

We actually match /.(>|\d+)/g, which means that the substring <> is interpreted as '<'.repeat('>'), resulting in an empty string.

JavaScript (ES8),  266  262 bytes

s=>` _2
 1
 -2
 7\\ 3^__^
 8\\  (oo)\\_7
 11(__)\\ 7)${q=`\\/\\
 15||<>`}-4w |
 15|| 5||`.replace(/.(>|\d+)/g,(s,n)=>n^1?s[i=0].repeat(n^2?n:w):a.map(s=>q[j='7019'[!i++*2|!a[i]]]+s.padEnd(w)+q[-~j]).join`
`,a=s.split`
`.map(w=s=>` ${w>(v=s.length+2)?0:w=v,s} `))

Try it online!

How?

We first split the input string into lines surrounded with 2 spaces and compute the width w of the biggest line (including said spaces):

a = s.split`\n`.map(w = s => ` ${w > (v = s.length + 2) ? 0 : w = v, s} `)

We define q as:

"\/\[LF] 15||<>"
 ^^        ^ ^
 01        7 9

This string is used simultaneously as a substring of our template (except the <> part which is removed) and as a lookup string for the surrounding characters of the text bubble, using the marked positions above.

Each number n in the template string is interpreted as follows:

• 2: repeat the preceding character w times

• 1: insert the text bubble:

  a.map(s =>
    q[j = '7019'[!i++ * 2 | !a[i]]] +
    s.padEnd(w) +
    q[-~j]
  )
  .join`\n`
  

• anything else: repeat the preceding character n times

We actually match /.(>|\d+)/g, which means that the substring <> is interpreted as '<'.repeat('>'), resulting in an empty string.

JavaScript (ES8),  266  263 bytes

s=>` _2
 1
 -2
 7\\ 3^__^
 8\\  (oo)\\_7
 11(__)\\ 7)${q=`\\/\\
 15||<>`}-4w |
 15|| 5||`.replace(/.(>|\d+)/g,(s,n)=>n^1?s[i=0].repeat(n^2?n:w):a.map(s=>q[j=i++?a[i]?7:0:a[1]?1:9]+s.padEnd(w)+q[j+1]).join`
`,a=s.split`
`.map(w=s=>` ${w>(v=s.length+2)?0:w=v,s} `))

Try it online!

JavaScript (ES8),  266  263 bytes

s=>` _2
 1
 -2
 7\\ 3^__^
 8\\  (oo)\\_7
 11(__)\\ 7)${q=`\\/\\
 15||<>`}-4w |
 15|| 5||`.replace(/.(>|\d+)/g,(s,n)=>n^1?s[i=0].repeat(n^2?n:w):a.map(s=>q[j=i++?a[i]?7:0:a[1]?1:9]+s.padEnd(w)+q[j+1]).join`
`,a=s.split`
`.map(w=s=>` ${w>(v=s.length+2)?0:w=v,s} `))

Try it online!

How?

We first split the input string into lines surrounded with 2 spaces and compute the width w of the biggest line (including said spaces):

a = s.split`\n`.map(w = s => ` ${w > (v = s.length + 2) ? 0 : w = v, s} `)

We define q as:

"\/\[LF] 15||<>"
 ^^        ^ ^
 01        7 9

This string is used simultaneously as a substring of our template (except the <> part which is removed) and as a lookup string for the surrounding characters of the text bubble, using the marked positions above.

Each number n in the template string is interpreted as follows:

• 2: repeat the preceding character w times

• 1: insert the text bubble:

  a.map(s =>
    q[j = i++ ? a[i] ? 7 : 0 : a[1] ? 1 : 9] +
    s.padEnd(w) +
    q[j + 1]
  )
  .join`\n`
  

• anything else: repeat the preceding character n times

We actually match /.(>|\d+)/g, which means that the substring <> is interpreted as '<'.repeat('>'), resulting in an empty string.