Getting into something new can always be daunting at first, but the more you struggle through something, the more you learn, providing you're making progress. I hope these answers help you in your journey.
1) Song structure is one of the most important aspects after understanding what the beat of a song is. I was literally talking to one of my piano students about this tonight. There is a pattern to most songs at least when we're speaking about the pop music genre.
One thing to note is that pop music refers to a plethora of musical styles: hip-hop, electronica, country, rock. It's basically anything that's not world music (Indian music, classical chinese music, etc) or jazz. If you're going to be DJing it, it's pop.
Let's talk structure. Most songs have the following structure intro (4 or 8 bars), verse(8 or 16 bars), chorus(8 bars), verse, chorus, bridge(8 bars), chorus, chorus. There are many examples of songs that deviate from this pattern, but it's not really important. What's important is that you see the length of the sections. Choruses are usually 8 bars in length. I can say that pretty much with 95% surety. The question that you're probably asking right now is, "what in the world is a bar?". Glad you asked. A bar is simply a way for music to structure these things called beats. In almost all pop music there are 4 beats in each bar. So a chorus, for example, would have 32 beats since it is usually 8 bars long. What is a beat? Here's an easy way to think about a beat. If you were bobbing your head to a song, each "bob" equals 1 beat. If you "bob" your head 4 times, that's 1 bar's length worth of "bobbing".
As a DJ, you need to understand what your beat is for each song. The length of a beat is dependent on the speed (or tempo) of a song. If it's a club tune, you'll bob your head faster, which means that the beats occur more frequently together, which ultimately means that the tempo is faster. Tempo is measured in BPM (beats per minute). A typical club track tempo will be between 120 and 140BPM. I'll speak more about this when I answer part 2.
In conclusion, song structure allows you to understand where sections of the song change and to anticipate that change. You may want to mix one song into another song from the end of the first song's chorus into the beginning of the next song's chorus. This has to be planned initially, but as you see (and more importantly feel) how long sections of a song are, this will be become more intuitive.
2) Now that you understand what a beat is, the concept of beat matching is a little bit easier, but you're not out of the woods yet. I would highly suggest buying a little device called a metronome. The metronome is a device used to keep musicians playing at a specific speed, but it can be used for a DJ as well. The way you would use it would be to play a song and practise getting the metronome to tick to the beat of the song. You'd be beat-matching with the metronome at first.
Advanced software like virtual DJ, etc, have the ability to figure out the BPM of a song for you. As an aside, sometimes it can be wrong. The software may say that a song is 70BPM when it's really 140BPM. Not a huge deal since when it's wrong it's usually half or double the actual speed.
The overall goal in beat matching is to take two songs that have a similar BPM and mix them together. Let's say one song is 80BPM and the other one is 120BPM, that won't work because they are not close enough. Songs should be within + or -10 BPM of each other. Some people may say that the amount can be more and they're right, but you have to know what you're doing. For starters, take two songs that are as close as possible. Remember, the software will identify how fast a song is once you play the song once through it.
Once you find your songs, cue each song to the start of the chorus. Play the first song by itself. You'll need to count beats. Remember, choruses are almost always 32 beats (8 bars) long. Once it finishes, immediately start the 2nd track then stop the other song. You'll want to make sure that the crossfader is in the middle so that you can hear both songs together. Honestly, your best bet is to check out some YouTube videos on mixing, I'm sure there are a ton of them out there. It's always harder to describe things via text.
3) When you ask "are there different ways to beat match" the answer is not really. In terms of what sections between the two songs you decide to start each song at, then yes, that will vary. There's really two different ways to beat match: to beat match well or not so well. We've all heard DJs who have no idea what they're doing and we've heard DJs who are amazing. It takes practise and a little bit of guidance to become the latter.
All the best in your quest :) Let me know if there are any points that need further clarification. By the way, I learned how to DJ by learning how to produce music and playing piano...go figure.
See videos below from JCPedroza.