6 months old is a time of MAJOR changes in a baby. It tweaks them and is not easy for them either.
And yes, teething IS painful, for many babies. So comfort him. And yes, they can't sleep when in pain. Comfort him. Nothing is wrong with that. A baby STILL needs to "bond" with their Mom even now and at different ages. Not just as a newborn.
Some of these things are:
1) Growth-spurt. They get mega hungry at 6 months old. You need to feed on-demand, 24/7, day and night... so that their intake keeps up with their growing and developmental changes. EVERY 3 months in a baby, is a growth-spurt time.
2) Teething time
3) time of major developmental changes, cognitive changes and motor-skills changes.
4) Time of emerging "separation-anxiety" AND "object permanence." These are developmental based things... which you cannot just make it go away. The baby is simply acquiring new abilities... and these are some of it. Look it up online. Its natural and normal. It means... your baby is developing well and normally.
AND on top of it all... ALL of these things can be occurring at the SAME time for a baby. It does not just occur one thing at a time. So... naturally babies get all "crabby" too.
I HIGHLY recommend the book: "What to expect the first year"... which you can find pretty much anywhere and online.
For the 1st year of life, a baby's PRIMARY source of nutrition is from breastmilk or Formula... NOT solids and NOT other liquids. AND they need to be fed on-demand. This is a building-block time, for the 1st year which is crucial for a baby.
My kids, often even went through what is called "cluster feeding" in which they even fed every s.i.n.g.l.e hour. Because they were growing and changing mega lots. Its normal.
Keep in mind, its not easy for the baby at these junctures either. They don't do it on purpose. Just feed and comfort. And no, it is not easy. But these growth-spurts and changes and hitting milestones in a baby is a phase. LOTS more phases will come up. This is not the end... but it does tweak their sleep ability as well. Because, their body and minds are changing so much. It means they are developing well... they need more nutrition and intake at these times.
Feed him more in his bottle... since it seems that he will and can feed more.
ALSO feed on-demand. The Formula or the breastmilk should be given BEFORE solids... not after. And at any other time baby indicates hunger.
all the best,