These are important questions. Many people do not know what to do, or find the relevant halachot to be complicated, as they may appear counterintuitive.
The gemara[1] takes the matter of properly preparing the body for a respectable tefilla very seriously, and this includes cleanliness. Accordingly, if one recites Shemoneh Esrei at a time in which he will be unable to restrain his need to defecate for 72 minutes (the ruling regarding urination is unclear[2]), his tefilla is considered an abomination, is disqualified, and must be repeated.[3] Even when one can wait 72 minutes, it is proper for him to take care of his needs before davening if he feels any real urge to use the bathroom, even if this means that he will be unable to daven along with the congregation as a result.[4]
If one started tefilla or a section thereof when it was already forbidden for him to do so, and he then reconsiders, he must stop immediately, irrespective of where he is up to. However, if his situation “deteriorated” more quickly than expected, the question of what to do depends on the section of tefilla and the severity of the urge. During Shemoneh Esrei, when it is forbidden to move one’s feet under all but the most extenuating circumstances, he must continue until the end and then use the bathroom, even if he will miss Kedusha.[5] Only if one reaches the stage at which restraining himself is degrading would he be able to leave his place in the midst of Shemoneh Esrei.[6] Even in that case, if he is davening in public and walking out in the middle will cause significant embarrassment, he may choose to continue davening.[7]
During Kri’at Shema and its berachot, one may choose whether to go to the bathroom or continue the tefilla.[8] However, since he may not start Shemoneh Esrei in that state, it is best to pause at one of standard breaks[9] in that section.[10] During P’sukei D’Zimra, one may go to the bathroom between any of the psalms or before Barchu. He should not wait until after Barchu, as that begins the next section of tefilla.[11]
If one stopped properly, he can certainly continue from the place in tefilla at which he stopped. Even if he should have stopped earlier, any part of tefilla that he recited other than Shemoneh Esrei is considered valid after the fact,[12] allowing him to continue from where he stopped.[13] If, however, he spent more time in the bathroom than it takes him to recite the entire section that he is presently reciting, he would be required to return to the beginning of the section.[14]
When to recite Asher Yatzar after exiting from the bathroom depends on the place in tefilla that one has reached. If he was in the middle of P’sukei D’Zimra when he left to use the bathroom, he should optimally make the beracha at the first break between “paragraphs” of P’sukei D’Zimra.[15] If he used the bathroom during Kri’at Shema or its berachot, he should delay reciting Asher Yatzar until after Shemoneh Esrei[16] (assuming that he will not feel a new urge to go to the bathroom by the time he has the chance to recite it). During Shemoneh Esrei, one would certainly not be allowed to recite Asher Yatzar under any circumstances.