Will Bruce Banner have broken bones once he's transformed back into his human self?

No

A lot of this is covered over on Science Fiction & Fantasy...at least from a comics perspective which seems to be closely followed by the MCU..at least as far as this Hulk is concerned.

The two essential points are:

• Nothing happens to Banner when the Hulk is injured. The Hulk has arguably a better healing factor than Wolverine.

In Age of Ultron the Hulk loses a tooth but Banner is not affected.

• The Hulk isn't injured when Banner gets hurt as the transformation to the Hulk heals all injuries.

In The Avengers, Banner relates attempting suicide by shooting himself in the head. The only result..."the Other Guy spit it out".

I don't think the fall would have broken any bones but it may have slightly injured him maybe spraining his leg but I don't believe that's the point of this scene. But I think simply no, he wouldn't have broken bones.

Would he even be damaged in the first place?

You need to remember that if the fall would have hurt him/damaged him the Hulk would have transformed to try and stop him from damaging himself. In The Avengers (2012) Banner has talked about attempted suicide saying

"I put a bullet in my mouth and the other guy spit it out"

When the Banner is under stress, being threatened, or is feeling other extreme emotion he seems to show signs of turning into Hulk.

Bruce usually transforms into the Hulk because of rage (or other emotions but predominantly rage), or when is faced with a life-threatening situation. Something like a subconscious self defense tactic.

I imagine Banners transformations into the Hulk as a reflex reaction, imagine you touch a hot stove with your hand, your brain will sub-consciously remove your hand whether you like it or not. In the same way if Banner was falling a large height I believe he would have transformed.

With reference to the comics however since few forms of death are instantaneous (even 100 pounds of C-4 might not instantly vaporize you) it can be presumed that if Bruce Banner became enraged during the explosion, he could transform into the Hulk and that transformation could prevent his destruction.

Would his human physicality affect his Hulk physicality? (Credits to answers posted on this SF&F question)

In the comics: The Hulk's regenerative powers will heal his leg in the time it takes for you to read this sentence. If he didn't have such powers, he would not be able to withstand the brutal amounts of damage he suffers in most conflicts. In this image below, the Hulk has 80% of the flesh blasted from his body by Vector. He regrows it in seconds.

Damage done to Banner does not necessarily transfer to the Hulk either. Any injury Banner suffers is immediately healed by his transformation into the Hulk. He can even shake off effects such as transmutation into stone or glass. The Grey Gargoyle turns Banner into stone in Incredible Hulk #363. Normal humans converted into stone by the Gargoyle normally DIE!

In the Cinematic Universe:

If you remember, in Age of Ultron the Hulk loses a tooth and when transformed back into Banner the missing tooth doesn't affect the human form of Banner.

He pretty much still owns the super healing factor he has in the comics.

What is the point of this scene?

I think the point of the scene was to make people laugh, the fact that Banner was about to reveal his other self to Valkyrie only heightened the humor when he landed flat on his face.

The pause of him not transforming to Hulk builds up suspense and tension to make it even more awesome when he turns into Hulk and grabs Fenrir.

This scene is meant to be humorous and at the same time build up suspense for when the Hulk is about to enter!