There's no no need to use a formation service. Incorporating a company in the US is very easy and can be done online, even by foreigners. All you need is a Registered Agent and they start at $25/year (in Wyoming).
I'd also like to point out that forming a LLC as a foreigner is a bad idea, as that might make you personally liable for US taxes. Either form a C Corp outright or have your LLC taxed as a C Corp. Then just zero out the profits. Some extra forms to fill out, but still better than paying taxes.
Exactly. If you're outside the US, you can only be a contractor, even if you work full-time for your company. There are no caps on how high a contractor's fee can be - that's not the case with employee salaries.
There are more fancy options, such as having a second, foreign company owned by you be the contractor - if you want to avoid personal income taxes in whatever country you live in. Beware of CFC regulations though.
I'd also like to point out that forming a LLC as a foreigner is a bad idea, as that might make you personally liable for US taxes. Either form a C Corp outright or have your LLC taxed as a C Corp. Then just zero out the profits. Some extra forms to fill out, but still better than paying taxes.