I say leave that screen in place. The damper at the hood may help keep critters from coming directly into the house, but you really want to keep them out of the vent turbing and the T-top, as well. When you say "t-vent," I'm picturing something that looks like a roof jack for an attic vent with the opening and screening on the underside of the outer arms to the cross-bar on the tee. Around here, wasps & yellow jackets would be the biggest worry -- their nests could plug up your vent. Nasty to try to clean out. As for attaching the tubing, metal vent tubing comes in sections that plug together -- a slightly crimped end plugs into a slightly expanded end of the next section. The roof connection will be the same -- the top end slides inside the base of the t-vent. (May be a mite tricky getting it all to snap together with the roof unit already in place.) A couple of self-tapping screws through the vertical sides of the tee into the top ring of the tubing would be good if you can do that. The t vent may have screw holes already. Caulk would be okay -- but probably not ordinary vinyl painter's caulk because your local government code people might require a fire-rated caulk as you would with a chimney for a wood stove. It is readily available, just more expensive and more durable. Regardless, you also want to wrap and seal each duct joint with duct tape. (That's actually what duct tape is for.) For this application, you really want to use the aluminized tape instead of the "handyman's secret weapon" --- the common vinyl-cloth duct tape. The vinyl-cloth stuff just won't hold up very well in temperature swings you get in the attic and on the roof. If you have hot summers, that attic and underside of the roof will get very hot. Cloth duct tape softens and degrades in high heat. It also loses its grip in very cold weather. If you live in a place with cold winters, you also probably want to consider adding a second damper up by your t-vent. That mitgates the chilling effect of a cold column of air above your stove vent, particularly on windy days. This post was edited by JWVideo on Tue, Jan 28, 14 at 12:35

Good advice. I never install the surface-mounted types of vent hoods that were used in your installation for just that reason. Ice/water membrane will normally help protect against ice dam melt water intrusion, but with your hood being that high up it's probably above the membrane. Somewhat unrelated, yet related...a house I built two years ago, the owners called me up in late January with "a huge, huge, problem". They still have a blanket of snow on their roof, it's been there since the first snowfall in December. Meanwhile, all the neighbor's houses have had snow come, then melt off the roof, then more snow, then again another melt off to expose a clean roof. I drove them around to a few other houses in town that I've built, all my roofs are holding snow. Took them a while to comprehend that it means that the heat they're paying for is staying inside the thermal envelope of their house and not going, literally, "through the roof". The good news is you've identified the problem and it'll be easy to rectify. Good on you for identifying the problem.

You might check with your local planning department to see if there is some rule about distance between roof vents, although I'm going to guess that there probably isn't. Most furnace vents are taller chimney-style vents that vent combustion fumes upward. A proper roof jack for a Vent-A-Hood exhaust hood has a lower profile and will vent horizontally and slightly downward at the roof line, so there probably isn't any conflict. You can see the type of roof jack recommended by Vent-A-Hood at the link below. The roof jacks are in the second row. Be sure that you get one sized to match the size of your ductwork. Here is a link that might be useful: Vent-A-Hood Ducting Accessories

Since these are both exhaust flues and ducts, no clearances are required. Clearances are only required when an air intake vent, door or window opening is located near a flue exhaust, vent hood or dryer duct terminal. You do need to consult your building code office, however, because there has been a change in some mechanical codes this year that require the installation of 'make-up air' ducting when using an exhaust hood rated over 400cfm. If you do not already have a make-up air intake duct, your code may now require one, and if it doesn't, you should still consider adding make-up air anyway.