1. Would AI be more aggressive because of refusing Open Borders ?
2. Would it harm your relations to AI because of refusing Open Borders ?
In my experience one time in which it can be dangerous to refuse open border is when you're weak and stand between two AI with a bad relationship.
It might be a coincidence, and almost certainly the friendly AI would have turned against you eventually, but I've had a few games in which I refused passage across my territory to a friendly AI that had clearly prepared to attack the AI on the other side of me. The DoW came not much longer after that.
It can also come this way: AI 1 asks you to go to war against AI 2. You refuse. A few turns later AI 1 asks for OB. You refuse. The "friendly" AI 1 ends up denouncing you and/or using its army against you instead and you get something like "your weakness is my opportunity". Bribing AI 1 so they attack AI 2 can buy some time, but if they can't reach AI 2 without OB with you, the "white war" only delays the DoW on you.
Another situation where I've seen the AI possibly pissed off about a refusal to give OB is when they try to connect two cities and must use the roads on your territory to complete the connection. I've been DoW in situations like this, but there were already border tensions and it's hard to tell if the refusal to give OB played any role.
Otherwise I frequently (in fact most often) refuse OB to the AI without seemingly any consequence (they often offer a DoF or agree to renew one despite that. As OB is a kind of stepping stone on the way to DoF, I guess it might delay a DoF from coming, waiting for another positive diplo modifier to replace that, like having traded, or sending TR).
Apparently the patch was supposed to make the AI more cautious about selling open borders to players with high tourism, but I haven't seen must results from that in my post patch games so far. They seem as happy as before to sell you a Cultural Victory for 1 gpt while you block theirs.
On some maps it can be highly beneficial to refuse OB but buy them from the AI. Before Astronomy on water maps this can block the AI from exploring parts or even whole areas of the map. That can be useful if you intend to wipe a nearby civ before the others discover its existence, or to increase your chance to be the one able to found the WC.
Is it possible to have trade routes without Open Borders ?
TR ignore borders. Personally I'd like them to split "Open Borders" into more specific agreements: one would allowing a foreign trade concession in your territory, allowing trade routes between two civs. A second type of agreement would allow religious units access to your territory (you could thus incite a civ to send you a GP or missionaries by giving it to them). Another would allow workers and scouts passage. The final type would allow military units the right to cross or use your territory, including stationing an aircraft in your cities. It would give a diplo hit with your neighbor that gets attacked because of that. I would also give the possibility to disregard closed borders, with attrition for the units entering borders without permission, and a diplo hit increasing every turns this lasts, leading to denouncement and possibly war.