Would Triple DES-X with 7 keys be much slower than standard Triple DES?
Well, there's a fairly obvious CPA attack that uses circa $2^{34}$ or so chosen plaintexts, and about $2^{120}$ trial DES operations. Because of that, the cryptographical strength is not significantly higher than standard 3DES.
As for code size and memory size, it'd certainly be larger than standard 3DES (you do more, and have more keying material lying around) - but it's hard to make a guess as to how much.
The scheme would likely be somewhat slower than 3DES. For 3DES, you can omit the intermediate IP/FP permutations between DES operations. In the scheme you've defined you would have to have them in there, and you would have to perform the columnar transposition as well. Now, it looks plausible that you could combine the FP/transpose/IP between DES operations into a single operation, which would reduce the cost somewhat, but it still wouldn't be as cheap as omitting them entirely.
On the plus side, this is effectively a 128 bit block cipher, so you do avoid the vulnerabilities that come with a 64 bit block size.