Skeetandtrap is not all one word, it is not one game but two rather different sports. Each calls for different gear if you want to do well at it. Specialization and concentration on one if you want to do REALLY well at it.
Skeet is a short range game, you need not shoot at a target over 22 yards away. Yes, there is a Skeet choke, just enough constriction to round out the pattern, although you can do ok with cylinder or improved cylinder.
Trap is shot at departing, rising targets, at maybe 35 yards with normal timing from the 16 yard line. Modified to full chokes are used. Handicap trap is shot as a separate event from 18 to 27 yards behind the traphouse - 36 to 40 yard targets, full choke usual. Trap doubles is still a third event, auto or O/U needed.
Your 1100 with interchangeable chokes will do fine to start. Use the appropriate choke tube and, for trap, an attachable comb pad to raise your eye for the rising targets will help.
For the serious competitor, NSSA Skeet is shot in four gauges and you must have multiple guns, interchangeable O/U barrels, or insert tube sets for an O/U to compete.
ATA trapguns have high combs and ribs for the rising targets, long barrels with plenty of choke for the distance.
Repeat, you can make a start with your 1100. I actually prefer mine, even though I have owned a variety of O/Us, pumps, and trap singles over the years.
I think it is easier for an accomplished skeet shooter to learn trap than vice versa. I started in trap and have never done as well in skeet; but have seen folks coming the other way make the change easily.
Your specific questions
1. As I say, your present gun is fine at least to start and learn the ropes.
2. You must have glasses for protection. There are all sorts of tints supposed to increase contrast and make the targets easier to see. Don't sink a lot of money until you have some experience.
3. I wear earplugs to shoot a shotgun, never found muffs that would not bump the stock with my build.
4. I carry shells in a bag on a belt, I do not want a variable weight of shells supported from my shoulder with a vest. A lot of shooters wear a vest for the shoulder padding but carry shells in a belt pouch for that reason.
5. Chokes covered above.
Dave McCracken on the shotgun forum here has a lot of experience with new shooters and can help you out even more.
