Are you on CM7?

http://androidforums.com/triumph-all-things-root/470282-rom-cm7-tg-reloaded-final-03-25-a.html

 

It seem like everyone with a triumph is having different problems with the triumph, it seems like it depends on when your triumph was made. usually if your problem is software related it can be fix by installing the right rom. for example ur saying your phone is running slow and unresponsive mine was running alittle slow on the stock rom. but when i flashed cm7 it became very fast and very touch responsive, not saying if you flash a new rom you will have the same outcome but stock just wasn't for me. but flashing a new rom comes with good and bad news. Just if your going to think about trying new roms read up on them before doing anything with them and make a nandroid backup. as for the BT haven't really used it on a lot of things mostly to connect my ps3 controller to it which works all the time. i did read that people do have trouble with the triumph BT connecting to car BT.
If you are thinking about rooting its fairly simple take about a minute with this you can root and unroot.
http://androidforums.com/triumph-all-things-root/455603-guide-rooting-your-motorola-triumph.html

 

Regarding your Bluetooth issue, try setting it to discoverable with no timeout and see if that changes anything.

 

im going to assume your running stock, with a bunch of apps

the other post is correct about the Discovery, your device descovery time is how long the device is visable to BT devices, higher time = longer visable to other devices, also BT shutting off, your phone may have a setting or there may even be a built in one with stock where its possibly killing the bt when it tries to go into sleep mode, lockscreen mode, changing some settings may help here


the slow touchscreen response im going to guess thats from an app, anytime i have an app that uses internet and always updates and things like that, like Facebook, its remain open in the cache, Programs being in the cache isnt bad, nor does it mean its eating ur memory, but a app like facebook if it ramains active while in the cache will, cutting ur phones memory, for me anyway to 20mb, 20mb or lower on my phone and touch screen seems lagged, force stopping facebook and browser if thats open tends to fix it for me

 

If it is like wifi it should be in the bluetooth area press menu and then options or something like that...

settings>application>all> then scroll to the app and click it then click force stop.

 

People are over looking the android market because its not called that anymore since you just flashed the gapps file it was labeled market until it undated to play store. so look for an app called play store. a lot of people i know keeps freaking out that they must have deleted the market when they don't know its now play store.

Well if you can find your market lol them download gmail.

 

Settings -> Wireless & Networks -> Bluetooth Settings