I'll go directly to your questions;
1) All standard treatments for chlamydia are virtually 100% effective. You don't say your partner's treatment, but if she also was treated with doxycycline (and took the full presecription) you have no worries about continued infection. However, it is recommended that all persons with chlamydia be restested 3-4 months later, just to be safe.
2) You already have had HPV, regardless of your partner's dysplasia. Everybody who has has several partners in their life, even if limited to long-term relationships, has had HPV; with few exceptions, we all get it. There is no way to know whether you might have been infected again from your current partner's dysplasia, but it doesn't matter. If you develop warts in the next few weeks, of course see a provider for evaluation and treatment. Otherwise don't worry about it.
3) Just like you, your new ex-partner probably has had HPV already. There is no reason for you to use condoms at this point. However, it would make sense for your partner to get the HPV vaccine, which is recommended for all sexually active women up to age 26 (and probably will be convered by her health insurance).
Read up on HPV. Start with the information provided by CDC (www.cdc.gov/std), the American Social Health Association (www.ashastd.org), and Public Health - Seattle & King County (www.metrokc.gov/health/apu/std/).
Good luck-- HHH, MD