Is it possible to win in MTG when you don't have enough cards in your library to complete your draw phase?

One way it matters is in multiplayer games. If Alice has one card in her library and has to draw four cards, and Bob has something that triggers based on how many cards are drawn, then it's important that Alice draw the card before she leaves the game.

The reason why the rule 104.3c is worded this way is because some effects instruct the player to do something after they have drawn the cards which might change the outcome of the game. It is a pretty rare scenario, but consider these:

You cast Blast of Genius with less than 3 cards in your library. You still draw as many cards as you can, then discard a card and deal damage equal to the discarded card's CMC. If your chosen target was your opponent and the damage was enough to bring them to 0 or less life, the next time state based actions are checked (immediately after Blast of Genius finishes resolving), the game will find that both players have lost the game, ending the game in a draw.

You have less than 7 cards in your library and activate Jace, Wielder of Mysteries -8 abiity while he has exactly 8 loyalty, which means that he will be put into the graveyard before the ability resolves, so his static ability will not apply. You will draw as many cards as you can, and then, as part of the ability, win the game. If for some reason, you do not win the game (e.g. the opponent cast Angel's Grace in response), you will lose the game immediately after the ability finishes resolving because you attempted to draw from an empty library.

The first half of rule 104.3c is mostly relevant when a player attempts to draw multiple cards at once. Howling Mine places separate "draw 1" triggers on the stack. This means that the opponent will get an opportunity to respond to each Mine separately. For example, if they had 3 mines out and 3 cards in the library, all Lightning Bolt, they will be able to cast all the bolts before losing to the third mine trigger. Or if one of the cards was a Twiddle, they will get an opportunity to tap the last mine, negating the draw and continuing to their main phase.

Barring any card that explicitly change this, the start phase of any turn has three steps:

• Untap. Nothing can be played here. Tapped permanents controlled by the active player are untapped, and that's it.
• Upkeep. "At the beginning of [your / each player's] upkeep" triggered abilities trigger and are put on the stack. Players gain priority and may play instants, activated abilities and cards with flash. Things on the stack resolve as normally as players pass priority.
• Draw. The active player draws a card, and "At the beginning of [your / each player's] draw step" triggered abilities trigger and are put on the stack. Players gain priority and may play instants, activated abilities and cards with flash. Things on the stack resolve as normally as players pass priority.

At any point before the triggered ability of any of the Howling Mines resolve at the last stage in the above list, if your opponent can play a spell or ability that kills you, they win.

Also, to clarify, "Priority" means "Now it's your turn to play cards and activate abilities. Say pass if you don't have anything to play." Passing priority is the basic mechanism that drives the game forwards: it is what allowed the next player to play their stuff, and if all players have passed priority in succession without playing anything, that is what makes the next thing on the stack resolve, or if the stack is empty makes the turn go on to the next step or phase.

This actually means that if the game (rather than someone passing) gave you priority (which means that the last thing that happened was that you played something, or it is your turn and either something had just finished resolving or a turn step has just begun), you don't get to say "Well, if you're not going to do anything right now, I will cast this card / activate this ability." If you have given your opponent a chance to cast things, and they have declined, the game moves a step forwards.

So you don't get a chance to "see if they have anything to cast" before you decide whether you want to cast anything. On the other hand, your opponent doesn't get a chance to cast anything before you have decided that you're done casting.

Sure it's possible, but you'd need to win at instant speed. An example: your opponent has 3 life left, and you're holding a Lightning Bolt. Then when the Howling Mine triggers but before it resolves (which would kill you), you point Lightning Bolt at him and kill him. You could also do this on your upkeep, before the draw phase.

Or are you asking why the rules says "he or she draws the remaining cards, and then loses the game the next time a player would receive priority" as opposed to simply "he or she loses the game the next time a player would receive priority"? That would be because it's possible to not lose the game even though you are supposed to lose. For example, if you are controlling Platinum Angel, then you don't lose for trying to draw from an empty library. In that case it makes sense for you to draw the remaining cards anyway.

No, players can't use the cards drawn in the draw step to win the game if the draw would make them lose.

Priority does not pass when a player does "anything at all". The points when players get priority are well defined:

117.3. Which player has priority is determined by the following rules:

117.3a The active player receives priority at the beginning of most steps and phases, after any turn-based actions (such as drawing a card during the draw step; see rule 703) have been dealt with and abilities that trigger at the beginning of that phase or step have been put on the stack. No player receives priority during the untap step. Players usually don’t get priority during the cleanup step (see rule 514.3).

117.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.

117.3c If a player has priority when they cast a spell, activate an ability, or take a special action, that player receives priority afterward.

117.3d If a player has priority and chooses not to take any actions, that player passes. If any mana is in that player’s mana pool, they announce what mana is there. Then the next player in turn order receives priority.

We care especially about 117.3a. In the draw step, players do indeed get priority after they've drawn their card. However, Howling Mine's card draw is not a separate event, it modifies the special card draw in the draw step, so that you draw 1+x instead of just 1 card. Only when all 1+x draws have been completed (or failed, as in your case) does the active player get priority:

504. Draw Step

504.1. First, the active player draws a card. This turn-based action doesn’t use the stack. [Your opponent would draw 4 here instead of just 1 because of the Howling Mines]

504.2. Second, the active player gets priority. (See rule 117, “Timing and Priority.”)

It should be noted that getting priority doesn't kill you, it's the state-based actions such as the one mentioned in 104.3c that do. They are checked first whenever a player would get priority. So it might look like getting priority kills you, but in fact, players lose the game just before they would get their next priority.

704.3. Whenever a player would get priority (see rule 117, “Timing and Priority”), the game checks for any of the listed conditions for state-based actions, then performs all applicable state-based actions simultaneously as a single event.[..]