here are MY answers to your questions:
Q1: Active speakers have built-in amplifiers. Passive speakers don't. So with active speakers, you don't need an amplifier anymore. With passive speakers, you do.
Q2: To my knowledge, the PS2 has a digital audio out. If it does, this means, you need a digital amplifier to be able to decode that digital signal and utilize the 5.1 audio.
Q3: Yes, there are. But this doesn't allow you to experience 5.1 audio. The only ways you can experience 5.1 audio is by either, digitally, using a digital amp with a digital source (i. e. DVD player), or analog, having a 5.1 channel amp and a source that has a built in digital decoder.
Q4: Yes, there are. But most 5.1 speaker system that has digital decoding properties are quite expensive. The most inexpensive 5.1 system that has digital decoding properties that I've seen so far are the ones from Kinyo (around 7K at Raon). It has 5 satellites and a sub and is Dolby Digital certified and has Pro logic capabilities for 2ch sources (i.e. audio cds, vcds). Maliit lang sya. There are also 5.1 systems that are inexpensive, but cannot decode digital signals, bale they're just 5 speakers na may amp lang and may sub, they rely on the on-board decoder of the dvd player, since your PS2 doesn't have this, this wouldn't be appropriate for you, unless you decide to buy an inexpensive dvd player(2.5K) that has a built in decoder.
Now, note that I never mentioned DTS in any of my answers. Let's get something clear. DTS is kinda like a brand of digital encoding and decoding. Dolby is another. Those two are kinda similar. There's Dolby Digital, and then there's DTS. Almost all the time, if a certain equiptment is Dolby certified, it is also DTS certified. But there are, of course, equiptments that can decode Dolby, but can't decode DTS. So, be careful about this. My friend has a xenon dvd player that has this problem. Now, DTS has become more popular these days than Dolby because DTS has lesser compression, resulting in a clearer, more natural sound. Dati kasi, Dolby was like the forerunner in this field, ngayon medyo nauuna na ang DTS, kaya medyo mas naging standard ang DTS kesa sa Dolby. Now, since you're on a budget, a 5.1 system like the Kinyo I suggested might be adequate for you already since you're using your PS2 as a player which doesn't have a built-in digital decoder. Plug and play na lang yan. The only reason why some enthusiasts opt for dvd players with built-in decoders is because of the fact that you don't need a digital amp to decode the 5.1 audio, since digital amps are quite expensive (12K up, amp pa lang yan). They just buy inexpensive amps that have 5.1 input (3.5K) and speakers to go with it/or complete 5.1 systems(with speakers and amps). Some just buy a Home-theater-in-a-box, like nextbase, may dvd player na, speakers, and amp na. These start from 8K up. Now, also remember the disadvantage of buying complete systems, they're much harder(costly, too) to upgrade and might have limited options, too. Buying your equiptment component by component is more ideal since mas madaling mag upgrade and mas mama-maximize mo ang use nya(i.e. buying an amp would enable you to use it for other sources like a vhs, tuner, cassette, vcd players, etc.). Now, don't worry about asking questions, because that's one of those times that you learn. Besides, ever heard of the expression, "teaching is learning again". TC, bro.
