Does finite rate of increase depend on mortality of individuals in a population?
*It's been several years since I've worked with similar equations. The following reply is based on memory, and if anyone has a firmer grasp of the materials, please modify or answer as you see fit.

I would assume λ is under ideal conditions or as an average of whatever species you're working with. Your modified lifetime reproductive output still leaves out many variables which can affect λ. By your own words it leaves out mutations and daughter cells that are dead-on-arrival. There could also be predation or conditions where the Mother-cell produces more than one Daughter-cell, resulting in R = 3 instead of 2 and your subsequent λ also being 3. UCLA Bioengineers have actually observed up to 5 daughter cells in cancer lines.

I'm not sure what you mean by "... or it depends on the natural situation?", but ultimately reality is always more complicated than basic equations usually describe. Either B or D can increase or decrease, depending on the species, environment, and how you define t.

Your equation still produces a population increase, just at a slower rate than your ideal conditions. If you're asking if there's a specific definition for "λ" referenced in your equation, I'm not aware of any specific definition for "λ" - it's not a Constant.

Your title question - does the rate of increase depend on mortality - is ultimately "Yes." If mortality is higher than the birth rate, then you will see a decrease of the population until extinction is reached if there isn't an equilibrium or rebound.

However, most populations change at an Exponential rate - NOT Geometric. To that end, you might be interested in the r/K Selection Theory and how they operate within Exponential equations (which are similar to your Geometric equations). In short, r-Selection is the reproductive strategy of producing extraordinary amounts of offspring with a very, very low survival rate. However, because so many are produced you only need a small overall percentage of survival to maintain or increase populations. The tapeworm I mentioned above is one of the stereotypical examples. K-Selection represents species that invest additional resources into raising a few offspring, usually just a handful that are carefully watched over and raised by the parent(s) until they are capable of reproducing themselves. Humans, elephants, most birds, and larger animals in general are K-selected.

For r-Selection your B and D will both be very high, and for K-Selection both B and D will be low. However, both will result in a positive λ simply because it's all about the ratios.