Is it possible to receive Luck Up collecting a coin

Without further informations about your current situation, I'll try to answer as thoroughly as possible with the occurrences that could prevent you from getting the Godhead achievement (I'll put in even those that don't apply to you for sake of completeness):

• Seeded runs: Whenever you input a seed you cannot get achievements in that run.
• Mods: using third party mods, even those that simply alter the monster sprites and have no interaction with gameplay, shuts the achievements down.
• Not playing in Hard Mode: this can be the most frustrating, since you can get to the end of a run and receive no achievement without noticing.
• Missing some of the bosses: to get Godhead you have to beat all of the following in hard mode: Mom's Heart, Isaac, Satan, ??? (a.k.a. the Blue Baby), The Lamb and the Boss Rush.
• Achievement bug: depending on your gaming platform and game version, you could have a bugged game. If somehow your game version is before v1.02, Godhead won't unlock. If this is the case, you should update your game and, sadly, you may have to re-unlock it.

These are, as far as I know, the only reasons why you cannot get Godhead. If these don't apply to you, please try to expand your question with further informations about your gaming platform, game version (pay attention that after v1.05 the version isn't showing anymore) and possible external interactions with your gaming experience.

Yes, it is possible, but very rare.

When rerolling your items, you do not lose any bonus health gained from items you previously held (though you do lose other stats, such as damage or speed). Since there are so many health-ups in the game, Chances are pretty good that rerolling will actually increase your health pool - you can see this simply by getting a lot of items and rerolling once.

Since you do not lose bonus health and instead gain more bonuses when rerolling, the only way to lose health on a reroll is if a new item in your collection decreases health. The only item I can think of that does this is Experimental Treatment, which randomly increases/decreases each of your stats. Occasionally ET will decease your health stat... however, when rerolling a lot of items at once, you might not see the decrease (again, since you might gain other items that increase your health at the same time). (Not to mention ET can't remove your last heart container.)

So in summary, the only way to lose health on a reroll is when you have more than 1 heart container and you reroll an item into an Experimental Treatment that randomly lowers your health, AND you don't get any other health-ups as part of the reroll. This situation is relatively rare, so in most cases you will not lose any health when rerolling.