Yavuz: Watch the arithmetic! I think the weight of your hose alone will exceed the tensile strength of any currently available hose material. If you pump water up the hose, how will you disassociate it into H2 and O2. If you pump up H2 and O2, two hoses will be required because they must not be allowed to come into contact with each other. Such equipment will add weight and increase the fuel you'll have to burn in each lift cabin to maintain position. You'll burn a lot of fuel just maintaining the cabins in position. Your operating costs will be huge and it will be difficult to smoothly handle cabin rocket engine failures and or replacements to avoid failures which may add fuel weight to your hose.
Can you do the arithmetic and present a quantitative comparison with ordinary rocket launching methods in terms of
payload efficiencies and costs?
Also, any such system has to lend itself to being developed and built preferably prior to construction of the production unit(s). How would you develop and test this system to assure that you had considered safety factors in your design to minimize failures and their consequences? Have you completed failure mode and effect analyses? For instance, thrust failures on any stage will immediately load the hose with the entire weight of that stage. How can disaster be avoided?
Antoniseb: If I may be allowed to help Yavuz with the problem you pointed out, he could program the guidance control system controlling his cabin thrusters to maintain the desired verticality of his hose...at an appreciable cost in efficiency.
Here, despite the title, is a discussion about the usability of a space elevator for Earth launches (or to avoid having to make them). Yavuz, you may find that discussion helpful. The advantage it would have over your system is that once you reached geosynchronous orbit, the effort to achieve and maintain payload orbit is much less difficult than for your system especially if you choose to use the system near either pole.
Either solar system colonization or interstellar travel and colonization will require a much more robust system than the one you have described. Reliability and maintainability are paramount for such a system.