A couple of things I've foudn with horns in general... the first is that a horn player will gravitate to the spot in the room where they hear their horn sound best... so what I would recommend to you is that instead of predetermining where your horn is going to go/be recorded you walk around the room like a horn player and find "the spot" where your horn sound really opens up in the room... my experience also tells me that those spots are generally near walls so the player can really feel the sound bounce off a wall but as always, YMMV.
When it comes to Trumpet my favorite mic is usually a Sony C-37a [early version with 6au6 tube] or the C-37p [which is a FET mic that runs on phantom power]. There is another version called the C-37 FET which runs on an internal 9v battery and the C-38 which I usually don't like as much on trumpet as the previously mentioned C-37's... I think that is mainly due to the different design of the head grill casing but I digress.
The bitch of the beast in my world is that I haven't owned a C-37 for years which has meant I have had to find an alternative solution... which after many trials and damn near as many errors I finally found the Royer SF-1.
The SF-1 captures the top end of a well played horn with the silk and opulence I want to hear from the horn but without the screeeetchy 6-7kHz 'ice pick in the forehead' shit that can also happen with condenser mics nor the 'honk' of a lot of dynamic mics. The other cool thing is that its pickup pattern is Figure 8 so I get a bit of room in with the trumpet sound... not enough to make the horn sound cloudy or distant but enough to give a sense of space and dare I say a bit more of what got the horn player to gravitate to the spot where they felt best playing the horn.
As far as mic-preamps, converters, storage, blah, blah, blah... that is another can of worms I don't feel like opening at the moment but for whatever it might be worth... I've found that the mic is the most important part of the chain [after the player's tone!!] when it comes to recording trumpet.
Best of luck with all you do.