Can [TQFN be soldered by hand]?
If you get a small hot plate and hot air, definitely yes. Others have described using a custom footprint, with a large plated through hole on the bottom, so they can use a fine tipped iron to solder first the IC pads (one at a time, tip in the corner between the PCB pad and the tiny bit of exposed IC pad), and finally the large center pad from the other side of the board.
I myself am exactly at that stage (and after that, onwards to BGA), having a cheap hot air station (see discussions in the
Other Equipment & Products subforum here for which one and what to look for), a tiny (for smaller than 2"×2" or 50mm×50mm boards) USB-C PD hot plate you can get off Banggood for a flash deal for under 15€ right now, and a
Pinecil USB-C PD soldering iron with full set of tips.
The existing soldering practice kits and DIY kits I've seen don't have interesting enough components – do get a few SMD LED projects or similar for a couple of USD/EUR each, if you haven't soldered any SMD components yet –, so I'll design my own, based on cheap microcontrollers easily programmed with open source tools (my workstations all use Linux). (I'm specifically looking at a CH32V305-based isolated UART/SPI adapter, designed to be partially populated for programming and testing as soon as the MCU and a couple of capacitors are soldered.)
You can find a lot of videos about the various soldering methods on Youtube, but be careful: there are a lot of garbage videos, so do check the comments, and skim a lot of videos, to see what works. I can tell that good flux, isopropyl alcohol (IPA), occasionally acetone, and an old toothbrush makes for a much nicer soldering job and results, for example!
which companies do [PCB manufacturing and assembly]
I use
EasyEDA or KiCAD to design my own PCBs. EasyEDA directly interfaces to
JLCPCB for PCB manufacturing and optionally SMD assembly, but for assembly you need to use their
parts catalog (which is very easy from EasyEDA, as it even shows the per-component price from JLCPCB or LCSC when selecting components). You can pick any such EasyEDA design and easily check the price by starting the ordering process, the price calculation is online and instant.
Unfortunately, JLCPCB does not seem to have any
max4885 in stock right now.
PCBWay is another, often used by for PCB manufacturing and optionally SMD assembly by us hobbyists, but I haven't personally used them yet (only because JLCPCB is so darned easy to use with EasyEDA).
The assembly is usually limited to components on one side of the board, and I do suggest soldering any through hole components yourself. I do plan to have components on both sides (just because!), but solder everything on at least one side myself.
As to the prices, you can get a set of ten PCBs up to 10cm×10cm/4"×4" for a few USD/EUR plus shipping costs. For SMD assembly, there are additional costs that vary, but as a rough idea, the setup etc. fees seem to be on the order of 25 USD/EUR for two/five/ten boards at JLCPCB and PCBWay, plus the part costs of course.