The best way to speed up combat in my experience is to make sure your players know their characters and actually pay attention to what is going on. When their turn comes up, they should already know exactly what they want to do.
If they are chit chatting or surfing the web and their turn comes up, and then they are like "What's happening? Who is that monster?" Or they wait until their turn comes up before they even begin to look over their sheet, then your combat will slow and nothing else will help until you address that. It also feeds on itself. When everyone takes 10 minutes, then other players start to check out and stop paying attention, so now their turn takes 10 minutes.
Once my group realized this was our problem, we got a handle on it. We started calling out players who weren't paying attention, or who just started looking at their sheet when their turn came up. It was draconian, but we actually started to time people at first. We no longer do that anymore. No need.
We went from 5 to 10 minutes per player turn to 30 seconds or less. Our 2 hour combats went down to about 30 minutes. It makes a HUGE difference.
As to your second point, yes you can bump up damage to help with grind and make combat more exciting and dangerous. +5 damage per tier per attack works pretty good for pre-MM3 monsters. MM3 and Monster Vault monsters already do good damage, so be careful about upping them as well.
When I first started upping damage, but players were like "Whoa! Stuff just got real!" Only they didn't say "stuff".
Another house rule I like is to bring in the escalation die from 13th Age. Essentially, you set a big d6 down in front of the DM screen starting with the 1 facing up in round two of the combat. Every round, you increase the number on the die by 1. This number is a bonus to hit that the players get (if you want to get really nasty, give the monsters the bonus too). Essentially, the players get +0 to hit in round one, but a cumulative +1 to hit for every subsequent round to a max of +6.
This helps reduce grind by increasing the odds of successful hit as combat goes on. It also encourages players to use their dailies later when they are more dramatically appropriate, instead of blowing their whole load in the first couple of rounds.