Is there an "issur" regarding not putting up a mezuzzah? Is there such thing as having 30 days until the issue kicks in? Also, can you say the bracha each time you put one up? Or, if two different people hang them up, can they each say the bracha?
I have information about everything but the theoretical "issur" question from Oholei Yeshurun, Vol. 2, Chapter 1: "Laws of Mezuzah", by R' Aaron Felder. Please note that while R' Felder is a talented and respected Posek who cites an impressive collection of sources, his is by no means the last word on the subject. If these questions come up practically, you should consult your Rabbi.
"Is there such thing as having 30 days until the issue kicks in?"
Here are the relevant passages from Section 3: "Laws Concerning the Time For Affixing the Mezuzah":
2) Immediately upon entering a purchased [as opposed to rented -IM] dwelling, the owner should affix a Mezuzah and recite the blessing.
4) A teneant who rented a dwelling outside of Eretz Yisroel for less than thirty days, is not required to affix a Mezuzah.
5) a) Upon entering a dwelling rented for thirty days or more, a blessing should be recited when affixing the Mezuzah.
... b) The Mezuzah may be affixed on the morning of the thirtieth day if one desires to wait until that time.
... c) If one wishes to affix the Mezuzah but delay recitation of the blessing until the thirtieth day, it should be touched before reciting the blessing. However, it is preferable to remove and reaffix the Mezuzah before reciting the blessing.
... d) The blessing should be said the day before the thirtieh day if it is a Shabbos or Yom Tov.
7) Even if one left the dwelling for a period of time during the first thirty days, a Mezuzah is still required by the thirtieth day.
Actually, the following might shed some light on your issur question:
13) One may remain in a dwelling that has no Mezuzah.
"Also, can you say the bracha each time you put one up?"
I'm not sure what case you're referring to. Do you mean each time you take down and put up the same one, or for each mezuzah in a houseful thereof?
If it's the former, it depends on the circumstances of the mezuzah's removal and replacement. There are a bunch of different cases discussed in Section 10: "Requirement of a Blessing Upon Affixing the Mezuzah" and Section 11: "Laws Pertaining to Conditions When a Blessing is Not Recited." If you amend your question to refer to a specific case, I'll quote the relevant ruling from these sections.
If it's the latter, here's the relevant material from Section 12 "Laws Concerning the Recitation of the Blessing":
4) a) When Mezuzus are being placed on several doorposts, one recites the blessing only once. This law applies only if one intended that the blessing be for all subsequent Mezuzos.
... b) The text of the blessing for many Mezuzos is the same as for affixing a single Mezuzah.
5) One should not converse from the time the blessing is recited until all the Mezuzos are affixed. If one did speak about a subject other than the mounting of the Mezuzos, the blessing must be recited prior to the placement of the remaining Mezuzos.
"Or, if two different people hang them up, can they each say the bracha?"
From the same section:
8) One who could not complete putting up all the Mezuzos, may appoint another person to complete the task. This person need not have been present when the blessing was recited. [Apparently, the second person does not recite a blessing in this case. -IM]
9) a) One individual may recite the blessing for several people in the same building. The one reciting the blessing should have the intention of including them, and those listening should have the intention of being included in the recitation.
... b) Those listening, should answer Boruch Hu U'voruch Shmo [I don't understand this ruling. -IM]. If Amen was not said, the blessing is not repeated.
I'll try and look this up for real and edit later... But here's what I can add from my recollection
1) You should try to mount main entrance mezuzot as soon as possible
2) There is a 30 grace period, I believe this relates to it becoming your established dwelling and not a temporary residence like a sukkah that is patur (exempt) from mezuzah
3) While not mounting a mezuza is nullifying a positive commandment I don't know of there is an "issur".
4) If you are putting more than one up at the same time, you may only make one Bracha. If you're putting them up with a substantial break in between then you would make another Bracha. One should not seek to do this since it is forbidden to make a "Bracha She'aino Tzricha" a Bracha that is not necessary.
5) It seems to me that if two different people are mounting them they can each make their own Bracha