Relationship between our microbiome and personalized nutrition
does the microbiome affect food metabolism?

Most definitely (and not surprisingly). The Arumugam paper [1] notes that

The drivers of [enterotype 1] seem to derive energy primarily from carbohydrates and proteins through fermentation, … because genes encoding enzymes involved in the degradation of these substrates (galactosidases, hexosaminidases, proteases) along with glycolysis and pentose phosphate pathways are enriched in this enterotype […]
Enterotype 2 … is enriched in Prevotella … and the co-occurring Desulfovibrio, which can act in synergy to degrade mucin glycoproteins present in the mucosal layer of the gut […]
Enterotype 3 is […] enriched in membrane transporters, mostly of sugars, indicating the efficient binding of mucin and its subsequent hydrolysis as well as uptake of the resulting simple sugars by these genera. […]
The enriched genera indicate that enterotypes use different routes to generate energy from fermentable substrates available in the colon, reminiscent of a potential specialization in ecological niches or guilds. In addition to the conversion of complex carbohydrates into absorbable substrates, the gut microbiota is also beneficial to the human host by producing vitamins. Although all the vitamin metabolism pathways are represented in all samples, enterotypes 1 and 2 were enriched in biosynthesis of different vitamins […]

[All emphasis mine.]

is the food that we eat affecting the microbiome?

Yes, just as certainly. I don’t have a publication handy but it should be obvious that our food influences our gut microbiome – in the extreme case, it can kill it (consider antibiotics side effects).
[1] Manimozhiyan Arumuga, Jeroen Raes &amp; al., Enterotypes of the human gut microbiome, Nature 473, 174–180, May 2011.