PC version

Steam update history

The first Plants vs. Zombies game is not getting active updates, as far as i'm aware.

Looking at the Steam update history, the last update was in 2010, announcing its release on XBLA.

I could not find any changelog for patches or other updates.

Why would it be updated?

From this Wikpedia article:

PopCap Games and its assets were bought by EA on July 12, 2011, for 750 million dollars. Fifty employees were laid off in the Seattle studio of PopCap Games on August 21, 2012, to mark a switch of focus to mobile and social gaming.

Why would EA go out of their way to Update a 10+ years old PC game, after they "switched their focus to mobile gaming"?

Buying the PC version is a one-time payment and the mobile version has both ads and in-game payments. There is not much monetary gain involved in updating the PC version.

It looks to me like the PC version is (by your definition) "dead", since there is a lack of updates.

Mobile version

Plants vs. Zombies FREE (Android) was last updated on January 19, 2021

Greetings zombie zappers! We’ve made some behind-the-scenes improvements to enhance your zombie-battling fun.

I'm not sure what those "behind-the-scenes improvements" are, but it is an update, isn't it?

I am trying to figure out if any updates are happening or will happen

The answer is "Yes, it is still getting updates", but that is most likely not the answer you were looking for.

The official EA website for PvZ1 (mobile) gives a HTTP 301 redirect to the official PvZ2 website without any explanation.

This is not a fact, but more of an opinion, but it looks (to me) like EA has stopped actively working on the first Plants vs. Zombies and focuses more on their newer installments in the Plants vs. Zombies franchise as seen by the redirect to PvZ2.

Those "behind-the-scenes improvements" are most likely just minor bug-fixes.