You played that partially correctly. Each player does get their own trigger to get a Lightning Rager token. However, your opponents' Lightning Ragers can attack your teammate.
Rule 810.7 says
The Two-Headed Giant variant uses the combat rules for the shared team turns option (see rule 805.10). This is a change from previous rules.
Then rule 805.10b says
The whole team is affected, according to rule 805.8 as of the 28 August 2016 revision of the comprehensive rules:
805.8. If an effect gives a player an extra turn or adds a phase or step to that player’s turn, that player’s team takes the extra turn, phase, or step. If an effect causes a player to skip a step, phase, or turn, that player’s team does so. ...
No, the Pridemate's ability doesn't trigger when your teammate gains life, because while your shared life total goes up it is not you gaining life.
Two-Headed Giant FAQ:
How does damage, loss of life, and gaining life work with the single life total for the team?
Damage, loss of life, and gaining life happens to each player individually. The result is ...
In Two-Headed Giant you need 15 or more poison counters. See Comprehensive Rules.
704.5t In a Two-Headed Giant game, if a team has fifteen or more poison counters, that team loses the game. See rule 810, “Two-Headed
810.5. With the exception of life total and poison counters, a team’s resources (cards in hand, mana, and so on) are not ...
Dealing 10 damage to a single player will not kill a team that has 20 life. This is where you are going wrong:
the single player life is the (total life)/2
This might have been true years ago (more on that later), but by current rules, individual players in two-headed giant do not have life totals. You cannot divide the total life in half and assign that ...
You can search for any card from the combined pool.
From the Magic Tournament Rules for the Two-Headed Giant Limited format:
Cards not used in a team’s starting decks are considered a shared
sideboard by the two players that both players can access.
Comprehensive rules, 810.9:
[In a Two-Headed Giant game, d]amage, loss of life, and gaining life happen to each player individually. The result is applied to the team’s shared life total.
If Player A has a Dread on the battlefield, "you" refers to its controller, Player A109.5. Dread's ability will only trigger if player A is dealt damage. Attacking A'...
Not only are you allowed to share cards, you are as a team considered to have a single card pool that you build your decks from. Section 9.6 of the Tournament rules says
Two-Headed Giant Limited Rules
All the rules for Limited Tournaments (Section 7) apply, except as described below.
The DCI recommends that each team receive eight boosters per team for Two-...
If an effect causes a player to skip a step, phase, or turn, that player’s team does so.
Similarly, if an effect gives a player an extra turn or adds a phase or a step to that player’s turn, that player’s team takes the extra turn, phase, or step.
You only damage one player, so only one player will exile a card. You get to chose which player you damage in the Combat Damage Step.
810.7a Each team’s creatures attack the other team as a group [...]
810.7f As the combat damage step begins, the active team announces how each attacking creature will assign its combat damage. If an attacking creature would ...
Propaganda etc. apply whenever the enemy team attacks your team.
In 2HG games you cannot be attacked individually, you always attack or defend as a team, so all attacking creatures attack both players at once and Propaganda costs will have to be paid for those creatures that don't attack Planeswalkers.
810.7a Each team’s creatures attack the other team as a ...
That rule is using "manipulate" in the sense of "physically interact with". It's saying that you can't grab cards out of your teammate's hand or tap your teammate's permanents, for example. This does not prevent you from targeting or enchanting your teammate or their cards/permanents.
So, the way 2 headed giant works is each team shared the turn, but each player gets the same phases. So when a card says "at the beginning of each player's upkeep", both players on the team get the token at the same time because both players are taking their upkeep step (it just happens to be at the same time). You played that correctly.
Your opponents ...
I know that user3490 has the correct answer, but here is an answer from the comprehensive rule book (my preferred authoritative source for these sorts of questions, when available):
100.4c In limited play involving the Two-Headed Giant multiplayer variant, all cards in a team’s card pool but not in either player’s deck are in that team’s sideboard.
Many strategies get better or worse in Two Headed Giant. These are the same as what changes in regular sealed.
Alternate win conditions are typically stronger as their thresholds don't change with the number of players. Making one player run out of cards in their library will cause the team to lose. Similarly having one player win the game, say ...
If the primary player has moved forward to the draw step, then the trigger was missed. If the secondary player is the one that drew a card, then that player has drawn extra cards.
In either case, at regular REL, you should call a judge, explain what happened, and allow the judge to decide whether the trigger was missed or whether the game should be rolled ...