If I apply for a German blue card, how much would the validity of the blue card be?
Up to 4 years or in cases that you have a time restricted contract, the length of the contract plus 3 months. (§18(4) - Principle of skilled immigration; general provisions - AufenthG)
When I renew my non-EU passport from Germany, are there any procedures to be followed with respect to the blue card?
No, but you should report that you have a new passport to the Immigration Office.
During my non-EU passport renewal from Germany, I will not have my passport with me. But German law considers it a requirement to always have a passport for foreigners.
You must possess a recognised and valid passport (§3 - Passport requirement - AufenthG), which is not the same as always having it with you at all times. In most cases a country will cancel the old passport after issuing a new passport. If this is not the case, you should ask your Consulate for a written confirmation that they have your passport while it is being renewed. Send a copy of this to the Immigration Office.
If the German police randomly ask me to show my ID during this period of renewal, what would happen?
You would show your residence permit card, which within Germany, is a valid form of identification.
Lastly if the passport expires (because I fail to renew it on time), will my stay in Germany with an expired passport but valid blue card be considered illegal?
You are required to undertake reasonable measures to insure that you have a valid passport. Keeping some form of proof when you made the, timely, renewal application is a good idea. Should the process take longer than estimated (i.e. beyond your control), then keep the Immigration Office informed of the situation.