1. Pretty much all slot 1 carts have 100% compatibility.
2. Again, all carts can view videos and listen to music with a homebrew application called MoonShell.
3. Again, no slot 1 cart seems to vary very much in terms of battery use, unless you have a defective one (such as a few CycloDS carts).
4. Most carts CAN play games at 100% speed. The real issue is with the MicroSD card that you use with the flash cart. Highly recommended are Japanese Kingston MicroSD cards and class 6 MicroSDHC cards. Even if you get a slow one, most games play fine, save Castlevania, Tony Hawk games, and a LITTLE lag on fast action games.
5. Ummm define fast, most carts can get to the game within 5-10 seconds. Especially light/fast ones can do it in maybe 4. And you have carts that go directly to a list of games, set up nicely, and if you click fast enough, within 2 seconds.
6. To go on webpages you can use homebrew applications. But if you want the official browser, not only do you need the flash cart, but also a slot 2 cart (I recommend the EZ Flash 3 in 1, only $20, you get GBA, RAM (for web browser), and rumble).
7. I should point a post a few threads down which is a rant about users talking too much about price. For the extra features, high end carts such as CycloDS Evolution are WELL worth the price. I mean another $10, will it really kill you? It just isn't worth it to get a cheap $15 cart that could break your DS, or a $20 cart that you will grow to hate and want to upgrade later.
Anyways, you will most likely be pointed to:
1. CycloDS Evolution - costs more than $50, but well worth the price. Will deal with all your problems. It has features such as real time save (or more commonly known as save state), real time guide (being able to pull up a guide at any time), slow motion, etc. Their support team is amazing, and every reviewer I know that have dozens of carts personally use the CycloDS Evolution.
2. AceKard 2 - About $25 at DealExtreme, there have been defective batches, but apparently they have been fixed. It is pretty fast, and meets all your needs.
3. EDGE - Must more traditional, R4-ish cart. It is basically a CycloDS Evolution stripped of all its features, again about $25 at DealExtreme. I highly recommend it for beginners.
4. Supercard DS ONE - $28, has all the awesome features of the CycloDS Evolution, but a horribly slow menu, one of those carts that will take the full 10 seconds to get to a game. But really, it has a nice set of features, and a great price tag. As well, you can choose to use a cheaper, faster menu without the features, that will get you to the game faster than the EDGE or AceKard.