A novice biker in Manhattan has the life expectancy of a fruit fly. His parked bike even less. No chain or cable can resist a New York thief's cutters, or he'll use his tools to strip it. You'll return to just a frame and wheels (if you properly locked them, too). No seat, no handlebars, no pedals, no go.
For a pleasant ride Central Park can be nice. Just don't park the bike out of sight, never ride at night, and try to stay in busy areas. Bike-jackings do happen.
I definitely would not plan to use your bike for getting around to other parts of the city. Murderous traffic and the theft problem as I said, unless you can bring your bike indoors, which is what office bikers do. Instead, buy an unlimited transit pass for the length of time you'll be there. Good for buses & subways, the subways much improved from some years ago. Just avoid them at night.
As for gay bars, that's always a matter of personal taste, and what you plan to do, and if you're alone or with someone. One gay man's bar is another gay man's poison.
Bars I've personally visited: Stonewall as a gay ticket-punch, to say you were there. A big nothing as an actual working bar, so walk down the street to the Monster while you're there. Has dancing downstairs for a younger crowd (after 10 PM as I recall), your typical over-the-hill piano area upstairs, but also some cruising around the main bar. So-so overall.
Splash has been named, not bad for someone your age. Also Therapy, which I find kind of young-professional-tame, with casual seat groupings and plenty of women, pricey drinks, everyone engaging in loud, trendy talk. I think you really need to be with someone to make Therapy enjoyable.
The Eagle offers lots of action, plenty of leather boys and shirtless wonders running around. Just beware of hustlers, and it's not in a great area at night. At the opposite end of the scale, for the Park Avenue set, the Townhouse, one of my favorites.
Which is literally what it is, a brownstone with residential decor, and you might want to phone ahead to see what nights they may have door control. Minimum attire is professional-preppy, but business suits after the office are also common. Very discreet exterior with merely a brass wall plaque, although the limousines outside are a hint, not to mention the guys inside with their own bodyguards.
Also best visited with someone along (in fact, they may bar single men they don't know, I've always gone with someone). But surprisingly the prices not that out of line for New York, and I recall the martinis being less than at Therapy. Has a pianist, though I'm not sure what nights.
If you're a museum goer like me, these are my favorites: MOMA, Metropolitan, Natural History (with the planetarium), and the Frick Collection, to which my ex-BF belongs. There are more, but those alone could take over a week.
Guided tours can include Rockefeller Center, with a separate tour of Radio City Music Hall, the United Nations, Ellis Island & Statue of Liberty (you take a ferry). The major churches & cathedrals like St. Patrick's (across from Rockefeller Center), St. John the Divine (world's largest Gothic church), Trinity at Wall Street, the Stock Exchange, and Ground Zero of the WTC. All these I've done and more.
The Empire State Building is OK, but you'll wait 2 hours or more. I get to avoid it because I'm handicapped, but as someone noted above, the top of the GE Building (old RCA) is quicker and fun, too.
Hope you have a great trip! E-mail me directly if you have specific questions.