How is Bucky even alive?

There are several misconceptions in your question, so I'm just going to address the general question of "What happened to Bucky Barnes after he fell off the train?".

The first thing worth noting is that in Captain America: The First Avenger, Steve Rogers rescued Bucky from Hydra captivity after finding him having recently been experimented on by Dr. Arnim Zola. It is theorised that Dr. Zola was working on his own version of Dr. Abraham Erskine's Super Soldier Serum, and it is this that was being tested on Bucky when Steve found him.

We don't know a whole lot about the details of the experiment that was performed, but we do know that since that experiment, Bucky has survived a lot of things that should have simply killed him (such as falling from the train, working as an assassin for a few decades, being frozen and thawed multiple times, etc) so we can conclude that even if Dr. Zola's experiment didn't make Bucky as capable as Steve, it at least 'powered him up' somewhat.

Some point after falling off the train and losing his arm, Bucky was acquired by the new Hydra, lead by Dr. Zola (who was, by this point, working for SHIELD while creating his new Hydra from within). It is at this stage that Bucky underwent further experimentation, presumably including the addition of his mechanical arm and the beginning of the memory wipes that, judging from Captain America: The Winter Soldier he undergoes fairly frequently.

In short, Bucky is still alive because the experimentation done on him by Dr. Zola allowed him to survive his fall from the train - but unfortunately delivered him back into Dr. Zola's care, who turned him into The Winter Soldier.

First of all, it's important to note that the SSR didn't find Bucky's body. The Soviets found him, and gave him to HYDRA initially. Even though Zola was working for SHIELD using SHIELD resources doesn't mean SHIELD "gave him" Bucky's body to fix.

Secondly, at the time that Armin Zola was working on Bucky, he was not considered an "enemy". SHIELD probably genuinely believed he was working for them the whole time. Plus, the fact that HYDRA had already begun infiltrating SHIELD, means Zola likely had support from higher-ups to keep suspicion off of him.

It may sound a bit crazy to us, but what happened with Zola is very similar to what happened in real life at the end of WWII. Lots of scientists who were originally working with the Nazis in Germany "defected" to the US.

At the time, German scientists (especially physicists) were some of the most advanced and skilled in the world at the time, and the US really wanted their expertise -- and more importantly, they didn't want the Soviets to get them. Operation Paperclip was a very real military project designed to recruit as many foreign scientists as possible (I believe the number was well over 1500). In the vast majority of cases, these scientists were not particularly zealous supporters of their previous regime: they either worked for Hitler out of fear, our of desperation, or because it was where the really interesting work was happening.

With the MCU, this is basically how Zola came to work for SHIELD. After he was initially captured, Zola exhibited none of the usually traits of the HYDRA operatives the SSR had dealt with: he didn't kill himself, he gave up information freely, and he even rolled over on the Red Skull with little prompting. He then served a few years in prison, during which he was presumably a model prisoner.

When SHIELD started up, they would have needed scientists with experience dealing in the kind of unusual stuff that SHIELD deals with, and Zola was right there. Zola kept up the appearance of being a fully reformed SHIELD agent, much the way the other HYDRA moles did. SO it would make perfect sense that SHIELD would want Zola to come work for them, and give him access to the resources he needed to work on Bucky.

It doesn't make sense because you are making a false (and very strange) assumption. The SSR didn't find Bucky, the Russians did. He was then handed over to HYDRA where he was a prisoner for several years. The SSR had nothing to do with any of this.

Then later, when Arnim Zola was released because of Operation Paperclip, he resumed his experiments on Bucky Barnes. This is when he received his metal arm and started being an assassin.

In fact, the SSR (and later SHIELD) never knew Bucky was alive until the events of the movie Captain America: The Winter Soldier. In the movie, Black Widow says: "most of the intelligence community doesn't believe he exists."
His SHIELD file said he's "missing in action" because his body was never found.