Yes, I am convinced that the metaphors we use can greatly affect how we think. For example, on the topic of elections, we have many metaphors for winning and losing, for combat, for racing...but we have no metaphors that say "the election was so close that it was a fluke which candidate was able to take office and they got virtually the same number of votes." So we THINK of elections in black and white, win lose terms.
In some sense yes, but in a deep sense no. So perhaps it is better to phrase the question more specifically relating to specific components of languages and to specific behaviours or cognitive domains.
So, if I live in a culture where a particular social practice is ingrained and where people's discourses about those practices reflect the dominant meanings in society, then yes, I guess that one could claim to be influenced by the language in some sense. (e.g. in a christian community, one might call each other "brother" or "sister" and have metaphors like "lord willing" and "By the grace of god" and "goodbye" etc. and in such a community, it might not be a surprise if all this language exerts an influence on people, say, to follow the precepts of the faith). But this strikes me as being a not very deep observation.
Also note that the exact opposite could also be true: e.g. the fact that in many English societies people swear "oh my gawd" or "damn it" doesn't necesarily correlate with higher levels of belief.
When we look at the language structure itself, I think that if there is such an influence, it is a very small effect and probably limited to lexical items and idiomatic expressions if that. But language is much bigger than just the words we use: it has particular structure to it. As far as I know there are no studies that show that speakers of SVO languages think diffferently to speakers of SOV languages; or that analytic languages affect thinking differently to agglutinating languages; that unergative languages make one think differently to unaccusative languages. Conversely, I don't know of any studies that show that a particular domain of cognition (say Mathematics or Chess) is unduly influenced by a particular language (e.g. a claim that the case structure of the Russian language activates the same neural pathways as the King's Gambit.).
There is the added issue that these kinds of claims could be construed in ethnocentric and xenophobic ways which could legitimize unequal power relations. Wasn't this part of what made the debate between Daniel Everett and linguists so acrimonious? Part of the issue was that Everett was making a claim of this sort, that the particular structure of the Piraha language affected the cultural (and maybe cognitive) capacities of its speakers. (http://www1.icsi.berkeley.edu/~kay/Everett.CA.Piraha.pdf). So apart from the linguistic and scientific facts, there was an ideological dimension to the debate.
Thank you very much for your interesting answer and comments. Thank you also for the examples that you have provided. I agree with what you have mentioned. The answer to this question is still complicated. The problem is that there is more involved than just language and thought; there is also culture. Our culture - the traditions, habits, lifestyle, and so on that we pick up from the people we live and interact with - shapes the way we think, and also shapes the way we talk.