[RPG] Does a character need to add Weapon Specialization bonus damage when shooting a needle full of medicine at an ally

The +10 damage bonus does not cascade, but it does apply to each attack.

While there are often time/speed benefits to rolling attacks in groups¹ (some people like to use different colored dice), you still consider each attack separately. An 11th level fighter using Extra Attack will roll three separate D20s, evaluating hit or miss for each of them. After each hit, roll damage - any attack rolls you took the -5 penalty on get the +10 damage.

Compare the Great Weapon Master Feat's benefit...

Before you make a melee attack with a heavy weapon that you are proficient with, you can choose to take a -5 penalty to the attack roll. If the attack hits, you add +10 to the attack's damage.

...to a Barbarian's Reckless Attack feature:

When you make your first attack on your turn, you can decide to attack recklessly. Doing so gives you advantage on melee weapon attack rolls using Strength during this turn[...]

Note the emphasized text in each. Great Weapon Master is a choice made on each attack, while Reckless Attack is made before the first attack and applies to the whole turn.

So, you can turn GWM on or off for each attack. Say you roll a 18 on the die, then factor in whatever bonuses (ability score, proficiency, etc.) apply alongside the -5 from Great Weapon Master... and the DM tells you missed. You know that 18 is a good roll by itself, which means that the target has a very high AC. On further attacks in the same turn, you could choose not to suffer the penalty (and not to gain the damage) to increase your chances of hitting at all.

¹_{If you're going to roll all the attacks ahead of time for speed's sake, for fairness sake, you should declare if you're using GWM or not ahead of time. That way, you're not making a decision on information you don't actually have.}

All attacks with the weapon receive the bonus to attack and damage rolls.

The description for +1 weapons says (DMG, pg. 213):

You have a +1 bonus to attack and damage rolls made with this magic weapon.

This does not discriminate between attacks made with the Attack action and attacks made via a feature granting a bonus action, or any other distinctions we might make, so all attack and damage rolls receive this bonus.

If DDB is not adding the +1 to the bonus action attack, this is indeed a bug. You can work around this by customizing the bonus action attack in DDB to have a +1 bonus to attack and damage rolls.

I've added a Glaive +1 to my character Poley McPoleface, and I've run into the same problem. The workaround is easy enough, see this image: