Assuming your bike is specced out as per these pages: 9 speed Tiagra (which is presumably 4500 series), Shimano SL-R440 shifters, triple chainring, Tektro mini v-brake calipers.
Any Shimano 9 speed triple drop bar shifter/brake units will work with the 9 speed Tiagra derailleurs. 'Flight Deck' refers to Shimano's computer integration with buttons in the shifters and does not identify the units you have.
Mini V-brake calipers also work with road brake levers, unlike regular full length v-brake calipers.
Be aware that flat bar frames generally have a longer reach geometry than drop bar frames, as flat bars don't position the hands as far forward as drop bars. You'll need a short stem to compensate for that. A short stem will speed up the steering a bit as the bars need to move a smaller distance for a given angle of rotation of the fork and front wheel.
Replacing all the cables and housings is not technically necessary but you will be replacing a a big enough part of them that you might as well go head and replace everything. Retain old housings as you can use some of them to cut new sections to the proper length. There are plenty of good guides available such as this video from Park Tool. A special cutter is needed to cut housing and cable and alternative is a Dremel type tool with a metal cut-off wheel as this does not squash the housing (although it does melt the inner lining which need to be cleaned up).
Update based on additional info provided - OP has 105 shifters.
If you have the older pre 2006 5500 9 speed series I'd say it would be better to get something newer. Tiagra 4500 or Sora 3500.
If you have 5600 or 5700 series 10 speed triple shifters an option is to upgrade to a 10 speed cassette. The Tiagra 4500 derailleur will actually work in this setup as all Shimano road groups 10 speed and below used the same derailleur actuation ratio (ratio of cable length pulled to lateral cage movement).
If you have 11 speed shifters that trick will not work as the actuation ratio is different and most 11 speed cassettes will not fit on a 8/9/10 speed hub. If you determine the shifters are not compatible you can of course sell them on ebay to finance the purchase of an alternative.