After reading Darth Plaguis novel this clarifies a few things about them. Granted this shoud have been explained in the movies but whatever:
>What is their purpose:
They are economic company specialising trade in the Outer Rim teritories which due to the lax regulations before TPM, inflated in size becoming effectivly a monoply by crushing competition which allows them free range to manipulate market prices which they do.
No only this but due to the corrupt nature of the Senate, they were able to buy out various planet's representation in there such as Felucia meaning they are suppost to represent their interests but basicly render them as a major political power.
When the whole Outer Rim situation became rife with inter-planet conflict thanks to Sheev the Republic considered taxing trade routes for companies like TF so they could use that money to finance paramilitaries in that region since the Republic can't raise an army of it's own.
>Why are they an army
They're not really. As stated the Republic can't raise an army due to the principles of the Ruusan Reformations 1000 years ago which rebuilt the Republic. Since the Outer Rim is strife with lawless bandits, pirates and Sheev's handiwork which has caused the death of Jedi and the assasination of the TF's directorate (which lead Gunray to lead the TF) this promted the various corperations to increase the size of their defenses such as armed capital ships and battle droids.
>What would they be doing
Bitching in the echo chamber of theSenate I suspect but a lot of this stuff got set in motion thanks to Sheev and Plageuis
>Why are they shooting people for palpy
They're not. They are shooting people for Darth Sidious who wants the Naboo to sign a treaty which would allow the TF to assume control of the plasma resources on the planet which made it a big target to begin with since Naboo is also a new player in galactic affairs.
The whole situation didn't really go as planed but the objectives were to create another dead jedi situation which weakens the order and damages Chancellor Velorum's reputation. Whatever happened Sheev got himself elected as Chancellor but the Sith discovered the Chosen One was on the scene.
>How do you set up a blockade in space
Gunray did ask something similar to this since he was concerned that such a effort would cost fuel and sources of income but since Sidious is affilated with Hego Damask (Plageuis) who is a wealthy Muun, he covers the costs.
As for phyiscally setting one up well they would encircle the planet and they did threaten anyone who tries to enter or leave with termination. But apparently they did not anticipate a single ship being able to get through.
I hope that's an autisic enough answer but I realise that a lot of this may just lead to more questions so tl;dr it was political fuckery by the Sith.