Is it permitted to Daven in a public place if there are women there?
If there are women that are Tznius, it is permitted as long as they are not in your view (provided this isn’t an official shul).
עיין חתם סופר בתשובותיו חו”מ סימן ק”צ.
If there are women that are not Tznius and you are in an open place or in the same room, you should turn your face away and not daven while they are directly in front of you even if you cannot see them.
If you are not in the same room (like in the tent outside 770 during Tishrei), it is permitted as long as they are not in your view.
ראה שאלה 5970 וש”נ. ובמחיצה – שוע”ר עה, ח.
Q. Is there a Halachic issue with Davening outside without a roof?
A. One may Daven in a secluded area without a roof.
שו”ע צ, ה. שוע”ר שם, ה. שע”ת שם, סק”א.
Q. In general: Are there situations when we are not Makpid on a Mechitzah by Davening (I’ve been in shuls where they started Maariv (Kadish and Barachu) after Shalosh Seudos, before putting back the Mechitzah)?
No, by Davening a Mechitzah is required.
ראה חת”ס שם. ועוד.