Assuming the HT is VHF/UHF, yes, the type of coax is starting to matter. For example RG-8X, a common HF grade coax, loses about 8dB/100'(33m) at 70cm. You are going to want something like LMR-400 (2.5dB/100' (33m)) and keep the run as short as practical.
After all, if you are losing 3dB (or more) in the coax and connectors (remember there is insertion loss for each connector), then you are at half power already. So, given the usual HT, that's down to 2.5W. Still plenty of juice to hit a local repeater, but you are using up the margin you might need to hit repeaters farther away or other folks on simplex.
What I have done is set a mag-mount with a spare 2m 5/8λ whip outside the window on a steel post and run the coax in through the window and use a connector to go from SO-239 to SMA-Female, I think. The coax was only 25', I think the total loss was less than 2dB.
You wouldn't have the freedom to wander around the house, but that wasn't one of my requirements because if I left the shack, all you'd hear would be the children screaming or something. After all, there is a reason shacks have doors :)
As for the coupling of the two antennas, I imagine it might work, but I think your loses would be so high as to not be practical if you need to get out with any power. Hard to saw canonically without more information and even then, it would probably come down to experimentation.
Reference for coax loses: http://www.therfc.com/attenrat.htm