Assuming you have one handed weapons the answers are the same for ranged or melee with the details changing.

1. Four although opponents could draw other attacks. That would be three from the Triple Attack feat (requires Double attack) in one of your weapons and an additional attack with an off-hand weapon.

2. It all depends on the allies and they kinds of benefits they can provide.

3. Triple Attack is a -10 penalty for ALL attacks. Using a second weapon to attack is also a -10 penalty on ALL attacks. The dual weapon penalties can be reduced with the Dual Weapon Mastery Feats I-III which drops the penalties to -5/-2/0. Some PrC classes have access to "Multiattack Proficiency - weapon type" talents which reduced the penalties for making multiple attacks 2 points for each level takes so taken five times it will reduce the -10 penalty to 0 for that given weapon type.

A character can pull off ∞ with perfect rolls (or as many nets as you have)
Thanks to @Ko_sct in the comments for this strategy

The Character

Any level 17 Wild Magic sorcerer can accomplish this using:

• Wild Magic Surge
• time stop
• the Dual Wielder feat

The Loop

1. Cast time stop
2. Trigger Wild Magic Surge
3. Roll a 1 on the d20
4. Roll a 59-60 on the d100
5. Throw nets throughout time stop until the last turn of time stop
6. Draw two nets (possible because of Dual Wielder) as part of your movement when necessary.
7. On the last turn of time stop repeat steps 1-4

As long as you continually roll 59-60 and never run out of nets, you can do this forever.

How many nets?

The highest carrying capacity a level 20 sorcerer can get is:

• A goliath, bugbear, orc, firbolg, centaur, or loxodon
• with 30 Strength (gained through Epic Boons)
• the High Magic Epic Boon
• having used a 9th level spell slot to cast enlarge/reduce becoming Large.

This lets you carry 1,800 pounds which comes to 590 nets in 20 sacks.

Of course if you can just pull from a pile of nets, then the maximum number of nets is arbitrarily large.

However, if you are interested in a strategy to throw nets a little bit faster, see my previous strategy below:

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The most a single character can throw is 37

The Character

The character, let's call her Celeste from here on out, that can throw the most nets is a level 20 character with the following classes:

• Sorcerer (Wild Magic) 17 ------ (for Wild Magic Surge, Flexible Casting, Quickened Spell, the Dual Wielder feat, and spells)
• Fighter 2 ------------------------ (for Action Surge)
• Cleric (War Domain) 1 --------- (for War Priest)

..., at least 20 Wisdom, and the Boon of High Magic

The Preparation

For Celeste's preparation round, she follows the following procedure:

1. Cast haste with the Quickened Spell Metamagic.
2. Roll a 1 on the d20 for Wild Magic Surge.
3. Roll 01-02 on the d100 for Wild Magic Surge.
4. Take the Ready action to throw a net at the start of the round.
5. Use the effect of haste to take the Use an Object action and draw your first two nets of many.

The Throwing

Now that Celeste has prepared, she throws her nets:

• First off, she throw the net she had readied, then her turn begins:

Normal Turn

1. Roll 81-82 on the d100 for Wild Magic Surge.
2. Throw a net. ---------------------------------------------------------------------- [total so far: 2]
3. Draw two nets as part of your movement.
4. Throw a net. ---------------------------------------------------------------------- [3]
5. Use War Priest to throw a net. ------------------------------------------------- [4]
6. Use the effect of haste to take the Use an Object action and draw two nets.
7. Use Action Surge.
8. Cast time stop rolling a 4 on the 1d4.

Turn One of Time Stop

9. Roll 81-82 on the d100 for Wild Magic Surge.
10. Throw a net. ---------------------------------------------------------------------- [5]
11. Cast ray of sickness with the Quickened Spell Metamagic on an empty space (use Bend Luck)
12. Roll 81-82 on the d100 for Wild Magic Surge to recover Bend Luck.
13. Throw a net. ---------------------------------------------------------------------- [6]
14. Draw two nets as part of your movement.
15. Throw a net at an empty space. ------------------------------------------------ [7]
16. Use the haste action to throw a net at an empty space. --------------------- [8]

Turn Two of Time Stop

17. Roll 81-82 on the d100 for Wild Magic Surge.
18. Take the Use an Object action to draw two nets.
19. Throw a net at an empty space. ------------------------------------------------ [9]
20. Use War Priest to throw a net at an empty space. --------------------------- [10]
21. Draw two nets as part of your movement.
22. Use the haste action to throw a net at an empty space. --------------------- [11]

Turn Three of Time Stop

23. Roll 81-82 on the d100 for Wild Magic Surge.
24. Throw a net. ---------------------------------------------------------------------- [12]
25. Cast ray of sickness with the Quickened Spell Metamagic on an empty space (use Bend Luck).
26. Roll 81-82 on the d100 for Wild Magic Surge to recover Bend Luck.
27. Take the Use an Object action to draw two nets.
28. Throw a net at an empty space. ------------------------------------------------ [13]
29. Use the haste action to throw a net at an empty space. --------------------- [14]
30. Draw two nets as part of your movement.

Turn Four of Time Stop

31. Repeat steps 9-11. --------------------------------------------------------------- [18]

Turn Five of First Time Stop

32. Roll 81-82 on the d100 for Wild Magic Surge.
33. Take the Use an Object action to draw two nets.
34. Use the haste action to throw a net at an empty space. --------------------- [19]
35. Use War Priest to throw a net at an empty space. --------------------------- [20]
36. Draw two nets as part of your movement.
37. Repeat steps 8-26. -------------------------------------------------------------- [34]

Turn Five of Second Time Stop

38. Roll 81-82 on the d100 for Wild Magic Surge.
39. Take the Use an Object action to draw two nets.
40. Throw a net at an empty space. ----------------------------------------------- [35]
41. Use War Priest to throw a net at an empty space. -------------------------- [36]
42. Draw your last net as part of your movement.
43. Use haste action to throw a net. ---------------------------------------------- [37]

The Probability

First of all, your GM has to allow you to roll on the Wild Magic Surge table (to recover Bend Luck) every time you cast ray of sickness, and to have the chance when you cast haste. This is already extremely unlikely. Then there are the actual rolls.

I changed the procedure a few times to try to get the best odds of this completing. The rolls necessary are listed below (keep in mind that a high level Wild Magic sorcerer can roll twice on the table and choose either result via Controlled Chaos | Thanks for the mention in the comments @Speedkat):

• 1 on the d20 one time
  • (1/20)
• 1 or 2 on the d100 one time
  • 1-(98/100)²
• 81 or 82 on the d100 nineteen times
  • (1-(98/100)²)¹⁹
• 4 on the d4 two times
  • (1/4)²

Multiplying these probablities comes to one chance in just over three nonillion (or 3,558,330,300,000,000,000,000,000,000,000 in numerical (rounded))

If we take the average number of times we roll 81-82 on the d100, then our probability drops considerably to one in 8,081, but we only get an average of 20.6732 nets

Or we can take the average result for all Wild Magic Surges dropping our probability to one in sixteen, but we only get an average of 20.001 nets

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¹ You have to target an empty space with ray of sickness and the nets during time stop so that the spell doesn't end:

This spell ends if one of the actions you use during this period, or any effects that you create during this period, affects a creature other than you or an object being worn or carried by someone other than you.

Attacking empty spaces is perfectly valid according to the Making an Attack section of the Basic Rules (emphasis mine):

1. Choose a target. Pick a target within your attack's range: a creature, an object, or a location.