From what I am told tube amps are much better for the HD800 than solid state
Not necessarily, it's just that some amps (which tend to be solid state than tube) have a tendency to either be totally flat, which doesn't do any favours for how the HD800 has peaks in the treble; or if they distort at your listening levels, they get sharper, which makes those peaks worse. By contrast some tube amps tend to roll off the highs a little bit and do a little bit more of that when pushed, and OTL amps tend to do more of that, while they remain an attractive option considering you can find some that put in a lot of clean power at 300ohms while also having a boost effect in the midrange due to their high output impedance (note that this is not always the effect, ie, they'll affect low impedance headphones differently, not to mention that OTL amps have less power into low impedance loads).
You can find some solid state amps that don't get as sharp when pushed, like Meier and Violectric. As for OTL amps, the Little Dot MkII bloated the bass on my HD600 and while not totally bloating the HD800, still made for rather less controlled fast bass notes compared to, say, the Darkvoice DV336se.
I have no idea what balanced means.
If you mean the amplifier topology, the most basic thing to know is that it puts a separate L- and R- on the circuit instead of a shared GND, so you really can't just use a male XLR (amp end) to female TRS (headphone jack end) converter, as this will fry the balanced amp. In most cases it allows the amp circuit to produce more power, but that doesn't mean single ended amps can't do the same thing - look at how the Lyr2 makes more power at 32ohms than the Jotunheim (although the Jotunheim has more power at 300ohms; and by now the Lyr3 has even more power, but you still get more of a drop in output at 300ohms relative to what it can put into a 16ohm load).
The real thing you have to look out for (apart from using the proper cable) is that balanced drive having basically four amp circuits means you have more parts that can screw up later on (not that all SE amps have fewer parts). That also includes the cable, because if the amp doesn't have an optional single ended output, then you'd basically not be able to use it until you get a new balanced cable.
You can go with OTL tube amps if you don't have any plans to get low impedance, lower sensitivity headphones.
I personally find the Lyr 3 very attractive, with maybe a topping D30 because apparently schiit dacs are quite bad.
The way USB drivers develop, any of such USB DAC that doesn't use generic drivers to run its receiver chip can potentially get rendered obsolete when a new Windows update comes and the custom drivers don't work and doesn't get updated. The more common problem I've seen on Schiit DACs tend to have something to do with the drivers.
Also a side note, I have very good onboard audio with Realtek1220. In my "Speakers Properties" on Windows under "Advanced", I have 32 bit, 192000 Hz as my default format sample rate, do I really need a dac, or is this good?
You'll need an amp for more clean power, and then in turn you'd
more likely (which of course is not "absolutely") get a cleaner signal into the amplifier using an external DAC, or a one-box DAC-HPamp (which also keeps the analogue signal path even shorter).
Not sure what my exact budget is but I wouldn't want to go past 1000$, and I'd rather keep it closer to 500$.
a. AudioGD NFB-11 or R2R-11
b. JDS Labs EL DAC or ODAC with Darkvoice DV336se; or WooAudio WA3
c. JDS Labs EL DAC or ODAC with Meier Jazz FF
d. Violectric V200 with optional DAC module
e. Meier DACcord FF and Classic FF
