Which namespace should I use as regards to searching?
You should use the Sitecore.Modules.dtSearch.Core namespace. Please view the sample in the /sitecore modules/web/dtsearch/samples folder.
According to the v4 documents, there are three parameters: search, count and offset. What do count and offset return?
Count is the parameter which defines how many documents should be returned as a result of search. Offset defines the number of the document which should occupy the first position among all the found documents. These parameters help you to implement paging on the search results page.
Is there any way to specify a point in the content tree where to begin the search (for instance, from a certain node through all of its children)?
You can index certain content tree and use it for search, but you cannot set the starting point. You can handle the result documents by the itemPath (scPath) attribute.
Is there a way to specify which templates to consider in the search, for instance: ignore all items which are not of one type of template? If there is, then is there a way to specify that a keyword is only valid if matched within a specific field in a template?
The dtSearch indexer indexes the front end pages, but not nodes. You can filter the result documents. From the result document, you can get the itemID (scID) and itemPath (scPath) attributes.