I have problem with my hosting service, regarding database connections limitation.

I'm listing 9 latest images, 9 latest albums in sidebar and 9 updated albums in footer.
I think, it is running (executing SQL) every page(even for bots also), what i want, instead of generating every time i want cache them for some time.

Please advice on this.

Have you tried the static_html_plugin?

Yes, my site is already running with "static_html_plugin".

That plugin caches everything as plain html, but of course only for Zenphoto theme pages and not if you call that stuff on the page of another system like Wordpress. For that there a cache plugins, too. Anyway, since you did not tell anything about your setup I can only guess.