Welcome BLACKMARKET,
Well you've got the same frustrations that I used to & still have!!
First off!!! Dealing with overseas makers/factories is supposed to be cheaper, but in the longrun you end up paying for it! I work in the apparrel industry & deal with dozens of overseas factories! And their bottom line is profit! The order has to make money for them, it's no different than domestic factories & shops here! Even though some
say they don't have minimums, someone there is punching a calculator deciding whether or not to take the order! Their reasoning is that the time your job will be on a press whether it's manual or automatic, they can use that time & material on a bigger order! What you should do is draw up a contract with the factory & agree to do X amount of "runs" with them! It all about repeat business with many overseas factories.
This is usually my first stop when trying to source manufacturers
http://screenprinters.net/links.php
If you're really serious about using Manufacturers outside of USA then you need to purchase this book, The Apparel Production Sourcebook. You can get it from this site
http://www.fashiondex.com
Now for your second question...
If the shirt label is screened on the inside then you won't be able to screen over it & have your company logo recognisable. Some shirt mills can give you generic shirts without the tags. Some manufacturers can actually provide that service. You're in NY? I use these guys for labels...
http://www.itclabels.com
Outside NY
http://www.hangtags.com/
Again volume will ulimately control how cheap/expensive the work will be!
As far as embroidery, I would continue to source this if I were you. You don't wantto skimp with embroidery! It's based on size, no. of colors & stitch count. You want a higher Stitch count for it to look decent.
You should research some new specialty inks for screenprinting that can simulate Embroidered thread!
Now I've said a mouthful!!!
Hope this helps...