Most surface mount part are easy, I prefer them cause I don't like drilling holes. For resistors/caps (0805 is my size of choice), I grab them with plain old forceps (I'm grounded, of course), position them, and touch one end with a fine tipped iron that has a tiny blob of solder on it. Assuming I didn't twitch at the wrong time, the other end gets a similar blob of solder.
IC's are only a bit tougher. Grab one lead with the forceps, position the part, and tack down a corner lead. Then tack down the opposite corner. If you use a decent amount of rosin flux, you can then sweep a heavily tinned iron across the leads, soldering an entire side at once (the flux plus the solder surface tension will prevent bridging; if you do get bridging desolder braid clears it right up).
Well... the SOIC ones easy at least. If you get into TSSOP or QSOP it's more fun.
44 for shipping?
That's absurd.
PCB is looking good. I'm impressed that you've needed so few vias. I think you could clear up some of the loopy paths by swapping which amp in the quad packages is doing what. Same with parallel port.
Your analog (power connector) and digital (p-port) ground: keep 'em separate. Star ground > ground loops. Same with the shunt inputs, don't tie it into board ground.
By the by, the tons of through hole stuff doesn't mean I can't make the board. It mostly means drilling your own holes. Oh, and no plated vias in home boards; components could need soldering both top and bottom.