If your HDTV has component input and doesn't crap itself over a 480i signal you can still get a nice picture out of a PS2. Not as clean as component is on a CRT but still every bit as good as if you were playing PS2 on a backward compatible PS3 over HDMI. The cable I use was like $15 from Gamestop and my PS2 looks pretty nice on the bigscreen.
Why Slimline over Fat though? Just because it's smaller and quieter? These days there are a few good options for playing pirated PS2 games. From best to worst I'd rank them
1. Off an internal hard drive or internal SD card
2. Off a burned DVD
3. Shared from your PC over the network
For option 1 you probably want to have a PS2 fat. There are ways to use an internal SD card with a Slimline but they require some tricky soldering. Using an SD card or any kind of drive through the USB ports sucks, don't bother with it. Your games will run like doing this through the USB ports, as they're way too slow. You 100% want to do this through the network / hard drive adapter if you go this route to get good performance.
This is a bit of a pain in the ass but I can give you some pointers on how to do it if you're interested. No soldering or anything like that needed for a PS2 Fat though. This one is best IMO since it speeds up load times vs reading from the DVD. This will require buying a Free McBoot memory card (about $15 to $20 on ebay last I checked) plus a network adapter (there is a place in town selling them for $10), plus another $20 to $30 worth of adapters if you do the SD card method (depending on how you want to transfer games to your SD card), or another $15 or so if you want to use a SATA hard drive. There is virtually no difference between hard drive vs a Class 10 SD card for read speeds.
For option 2 your best bet is buying a Slimline, as the FreeDVDBoot works on all Slimline PS2. It works on many Fat PS2 also, but is dependent on which firmware and DVD drive in a given Fat PS2, which you won't know just looking at it from the outside before buying. This doesn't require buying anything extra, as the hack comes from using a program to embed the hack into the DVD you burn when you burn a copy of a game to disc. This is easily the least hassle free method, and will run just like playing legit discs. So no speedup in load times like you'd get in option 1.
For option 3 you can use any PS2 other than an SCPH-90000 series Slimline, eg the last model of PS2 Slimline ever launched. This is what the SCPH-90000 series PS2 Slimline look like:
Any other PS2 Slimline is fine though. Eg the models that look like this with the glossy center stripe are fine:
For this option you'll need a Free McBoot memory card and if you buy a PS2 Fat you'll also need the official PS2 network adapter. Though if you want to do this method you should really go Slimline (other than SCPH-90000 series) since they have a built in ethernet adapter. Then you'd connect the PS2 to your router via an ethernet cable and share roms from your PC. This is the method I use right now and it works really well most of the time, though there are a few games you'll get stutter in full motion video cutscenes doing this. And if you're running a lot of crap on your PC at the time and low on available memory you'll notice stutters in FMV cutscenes. I have yet to run into a game that stutters at all in actual gameplay though out of the 50 or so I have played using this method.