According to Buddhism, everything that happens, happens for a reason, which is due to karma. How does Utu Niyama, Bija niyama fit into this?

I'm not sure I'd summarize it this way:

So, the important point is: not everything happens for a reason.

For a start, the sutta is talking explicitly about "everything a person feels or experiences" -- paṭisaṁvedeti -- maybe that's more precise as a topic than "everything".

Perhaps more importantly it doesn't say that some things "do not happen for a reason" -- I think it suggests that sometimes the reason is "the weather" and so on, just that the reason is not "past actions" i.e. kamma.

And the term it uses for "reason", or "is based" (i.e. "basis"), is samuṭṭhāna -- which I presume is related to paṭiccasamuppāda -- so there's that theory too (which perhaps you're familiar with and already accept), of the "twelve nidanas", which are not exactly kamma; and if you were to say, "ah but kamma is the basis of everything else", there's a different theory i.e. the root kilesas which are said to be the centre of that wheel.

And I think of kamma as personal -- I am the owner of my actions (kamma), heir to my actions, born of my actions. And I think that's a useful theory -- it encourages Right Effort instead of nihilism -- and perhaps it encourages you not to blame other people for "your problems".

But there's the Anattalakkhaṇasutta -- which I think argues that form, feeling, and so on are "not-self" because you can't "compel" (or 'control') them.

It also says in AN 4.77 that the results of kamma are "unthinkable".

In summary I think this leads to the doctrine of emptiness or sunyata. Kamma is a theory or view, a way of seeings things, like a lens or a perspective. And it's a useful one sometimes and (so) it would be wrong to say there's no such thing -- but it isn't the only "reason", the only theory about why things happen or about the 'basis' of everything.

In the Sivaka Sutta (SN36.21), the Buddha explains that not all experiences are caused by kamma. Sometimes, it's just the weather or something else that's not extraordinarily significant. Not everything happens for a reason.

So, Utu Niyama explains change of seasons. Bija Niyama explains bile, phlegm etc.

There's even harsh treatment from others, as a cause, which are unrelated to our kamma or natural laws.

And of course, there are consequences caused by kamma.

So, the important point is: not everything happens for a reason.

As he was sitting there, he said to the Blessed One, "Master Gotama, there are some brahmans & contemplatives who are of this doctrine, this view: Whatever an individual feels — pleasure, pain, neither-pleasure-nor-pain — is entirely caused by what was done before. Now what does Master Gotama say to that?"

[The Buddha:] "There are cases where some feelings arise based on bile. You yourself should know how some feelings arise based on bile. Even the world is agreed on how some feelings arise based on bile. So any brahmans & contemplatives who are of the doctrine & view that whatever an individual feels — pleasure, pain, neither-pleasure-nor-pain — is entirely caused by what was done before — slip past what they themselves know, slip past what is agreed on by the world. Therefore I say that those brahmans & contemplatives are wrong."

"There are cases where some feelings arise based on phlegm... based on internal winds... based on a combination of bodily humors... from the change of the seasons... from uneven care of the body... from harsh treatment... from the result of kamma. You yourself should know how some feelings arise from the result of kamma. Even the world is agreed on how some feelings arise from the result of kamma. So any brahmans & contemplatives who are of the doctrine & view that whatever an individual feels — pleasure, pain, neither pleasure-nor-pain — is entirely caused by what was done before — slip past what they themselves know, slip past what is agreed on by the world. Therefore I say that those brahmans & contemplatives are wrong."
Sivaka Sutta (SN36.21)