It should always load to the TTMenu screen rather than the NDS menu screen. You may have a bad DSTT -- it may not be making proper contact with your NDS. If so, you should return it and exchange it for a new one.
Could be, but not likely. Do you have problems with regular game cartridges not being recognized by the DSTT? If not, the problem is probably the DSTT rather than the NDS. Assuming you have the DSTT inserted fully into the NDS, it should always boot to the DSTT's TTMenu (or if there is a problem with your microSD card or you don't have the menu properly installed you'll get the DSTT error icon).
Others have had similar problems with some (not all) DSTT's. You may have just gotten one of the bad ones. If you can exchange it for a new one, you should. If not, there have been several suggested "fixes" posted to this thread (use them at your own risk though -- some can damage your cart beyond repair).
just slightly "bend" the contact part of your dstt from left to right, not up to down, be gentle, but it will show a visable "bend" this will fix it, but as mentioned before, do at your own risk, you can brake it, tho doubtfull as its cheap bendable plastic especially by the contacts