Is it possible that we combine some states of a Markov chain, like in this figure? (All non-zero states combined)
1) If yes, what are the new transition probabilities, i.e. p1 and p2 and p3 in the figure?
2) If yes, can we still combine states of the Markov chain if it is not finite, for example the process in the picture has infinitely many sates, and we want to combine all non-zero states?
3) what are the transition probabilities if it is a birth-and-death process (a CTMC)?
There are no "new transition probabilities" here since the new process is not Markov. Actually it has infinite memory since, for example, the transition $\tilde0\to0$ is impossible any even number of steps after the entrance into $\tilde0$ and possible any odd number of steps after the entrance into $\tilde0$.
The conditions to still get a Markov chain when one lumps some states of a Markov chain are well known. If the lumped process corresponds to the partition $\bigcup\limits_CC$ of the state space of the original Markov chain with transition probabilities $p$, then a sufficient condition is that, for every $x$ and $y$ in some $C$ and for every $D$ in the partition, $$ \sum_{z\in D}p(x,z)=\sum_{z\in D}p(y,z). $$ When this condition holds, the transition probabilities of the new process are $$ q(C,D)=\sum_{z\in D}p(x,z), $$ where $x$ can be any state in the class $C$.