Can a druid in animal form use Handle Animal

The Rules

The rules don't address this in much detail. Strictly speaking, the animal is an independent, intelligent creature and should have its own initiative. That means it's going to act on its own in the absence of a Handle Animal command. The DM gets to determine what that means.

As you noted, this is no problem if you get to go before the animal does. If the animal goes first, it can be annoying.

Why That Sucks

Most games I've played in (including the one I'm DMing right now) don't handle it that way. I find that players find the extra initiatives confusing, get unhappy when the DM has the animal do something before the player can react, and so on. You've got what is essentially a powerful class feature that you can command, except in this edge case at the start of combat where sometimes you can command it and sometimes you can't.

The people I play with love consistency, and that edge case is inconsistent.

Solving It

We solved that annoyance in two ways:

1. Tell the DM a "default" command that you're giving the companion while you're not in combat. As it's a free action you can just keep doing it while you're wandering around town/the dungeon/whatever and have it continuously in effect. Normal ones are Heel (The animal follows you closely, even to places where it normally wouldn’t go) and Defend (The animal defends you (or is ready to defend you if no threat is present), even without any command being given). Which one depends on if you want the animal to attack something automatically if you get attacked, or to just follow and wait for your command. This removes the inconsistency, as at least the DM now knows that 6 seconds ago you told the animal to Heel in the last round, and it should probably keep doing that until you get a chance to react.
2. House Rule - Animal Companions always start on the same initiative as their PC. Not strictly necessary, but as it makes things simple and consistent, the people I play with like it.

On teaching a wolf tricks

This falls under the description of Handle Animal.

To make a wolf ride-able, the creature must most likely be reared. Wolves are wild.

Rear a Wild Animal

To rear an animal means to raise a wild creature from infancy so that it becomes domesticated. A handler can rear as many as three creatures of the same kind at once.

A successfully domesticated animal can be taught tricks at the same time it’s being raised, or it can be taught as a domesticated animal later.

During this time, you also train it in combat riding. Succeed on a few checks (DC 17 to rear, DC 20 to train), and you have a now-domesticated wolf that knows the tricks of "attack, come, defend, down, guard, and heel".

Then you hug and play with the thing until it becomes your animal companion. This teaches it an additional trick, which could be any trick you want.

As for whether you can use this bonus trick to teach it a whole new purpose - no. A purpose is just a selection of tricks, combined into one Handle Animal check for convenience. It does not give a discount on the number of tricks required for the whole package, just the number of checks.

On riding a wolf

The questions of whether you take a -5 on riding checks and need an exotic saddle is something that isn't addressed in the game, at least not directly. You will have to ask your DM.

To me personally, wolves and riding dogs look very similar. We also know from real life how physically similar wolves are to large dogs. I can't imagine one being physically well suited to riding and the other not, or them requiring very different saddles. I'd say once you have successfully reared the wolf into domestication, you're good to go.