The difficulty of encounters (based on EXP) from DMG is highly overestimated.
From my experience, either I'm awful playing NPCs at combats or the guidelines on encounters are intended for a really safe world. I've run lots of "Deadly" encounters where at worst my party would deplet their resources, but hardly (even ONE PC) would die from it.
This is specially true for the Adjusted XP, mainly for hordes of CR-lower-than-1 monsters
Example with Spoilers from The Death House (one-shot adventure that is a hook to Curse of Strahd, also regarded one of the hardest published adventures as far as I'm aware) ahead:
Near the end, the players are put against a CR5 Shambling Mound as a bunch of level 2 players. For a 4-PC party of lvl 2 players, a deadly encounter is 800XP - a CR5 alone is 1800, which is almost the entire adventuring day (2400XP). By the time they get there, they usually have used some of their resources due to numerous previous combats. As far as I remember the adventure, they were supposed to just run from it. The Barbarian decided he did not want to and they fought it. After many rounds, the Barbarian dropped to 0 HP and the Mound went for the other PCs. Then they started to kite it (mound moves slower, even with Dash it still closes up slowly) and eventually defeated it. TL;DR: Four 2nd level adventurers can defeat a CR5 monster.
So, don't underestimate your players.
Even if they are defeated, it doesn't mean they should be killed
For the ambush, the Book even explicitly states that
In the unlikely event that the goblins defeat the adventurers, they leave them unconscious, loot the wagon, then head back to the Cragmaw hideout.
While not explicitly stated, the same can go for the Bugbear fight - he can trap them and want to sell them as slaves, leading to a quest to free themselves before they can attempt to fight again.
About the encounters you mentioned
- Yes, the ambush is hard for new players. It's specially hard if one player goes alone to check the horses. That said, goblins are dumb - so don't instantly target the party's Wizard or whoever is easier to kill. They are also lazy and hard to coordinate even under a leader, so by themselves they probably shouldn't be hitting the same target, while the players should be focus firing. The main hardship from this fight is the surprise and hiding, once the goblins are spotted, 7 HP under a 15 AC is quick-to-die.
- All the encounters inside the cave can be done in a way that the players have the surprise attack. As you mentioned in the Goblin Ambush, the surprise is an important factor. They should be able to take out one or two goblins early on (read: on surprise round) and make it as easy/hard as the Goblin Ambush, but without the goblins have easy spots to hide. On a side note, I don't see how six goblins turned to a 1250 adjusted XP encounter. If they were 6 regular goblins, this would be a 600 Adjusted XP encounter. I don't see how one of them having +5HP more than doubles that.
- Same as above: players can surprise the bugbear. Also, it's a Boss Fight, it's supposed to be hard. Also, the bug bear runs away (or tries to) as soon as you kill the Wolf, making it a lot easier - similar to the Young Green Dragon at the tower, which runs away at half HP.
About the second mentioned encounter
From your wording and numbers, it seems the math you did was:
6 goblins = 300 XP
1 Goblin Boss = 200 XP
500 * 2.5 = 1250 adjusted XP.
There are two confusions here: First, you are confusing the Goblin Leader Yeemik with a Goblin Boss - this is NOT intended. If he was a Goblin Boss, it would be highlighted as such. The Goblin Boss stats aren't even on the starter set. Yeemik is just a regular goblin with 12 HP instead of 7.
Second, there are a total of six goblins, meaning 5 + Yeemik, not 6 + Yeemik.
The adjusted XP for this encounter is around 600 XP (around because Yeemik's CR is not defined), which while still Deadly (>500), it's way lower to the 1250 XP you calculated initially.