When bentching, there is a harachaman for either yourself, or the person hosting you. (harachaman hu yevorech baal habayis hazeh, etc). During shalos seudos in shul, when the seuda is hosted by another person, it would be obvious we should say harachaman hu yevorech baal habayis etc. But when I would sponser the shalos seudos, what version should I say? Also, when eating out at a restaurant, which one do I say as well?

The Harachaman in bentching acts as a blessing which is always advisable. Therefore, one can say it even when one acts as the sponser of the meal. Rabbi Ro'i Margalit