Good ways to speed up a game of Ticket to Ride
It will speed up with a few additional plays. Honestly, I suggest starting with people getting more experience before modifying the game.

You asked about how your ideas will effect gameplay, so I'll tackle that, having been involved in balancing a lot of games. The game was playtested hundreds of times to reach the balance it's at now, so some things to consider:


  keep score on paper


You'll lose the easy checking to see who's ahead and who's behind, which should affect your play. On paper you'd either have to do without or keep asking.


  play the game when each player has 4 less trains


If I were playing I'd be much more likely to dump the long tickets because it's not at all uncommon that they're completed in the last couple of turns of the game, especially with 4 or 5 players.


  start the game with 7 cards


Probably wouldn't cause any problems.


  get a free route at the beginning


Tricky. They'd have to be the same length (or same maximum length), and if it's long it will help people with western tickets much more than eastern ones. It would also change the early game quite a bit, where people traditionally build up a pretty large hand of cards before laying down, so as to not give anything away to others about how they could block you. It will likely hurt ticket completion a bit.


  You draw 3 cards if you choose to draw train cards. Locomotives still cost 2 normal draws.


I think this will help longer routes more than shorter ones, again with the western/eastern division. Each card draw will be slower (draw, fill, draw, fill, draw), but it would still likely be a speed-up. The bigger problem is that drawing cards will help your game much more than playing them, so it will again favor longer routes, making the already-somewhat-imbalanced longer routes much easier to complete, so the luck of the ticket draw will have an even stronger effect on what many consider an existing flaw (the 1910 ticket replacement add-on is designed to fix it).


  For each turn you draw 2 cards AND get the opportunity to claim a route.


This won't speed the game up as much as you'd think: the cleverness in the current system is that after you draw cards you have the whole rest of the round to think about what you'll do with them. With this change after drawing players will have to think through what they'll lay down while everyone waits. 

If they play first and draw second, they'll still need to weigh the cards on display with what route to claim, knowing they're about to draw from them. With the regular game there's no point in thinking about them because they'll likely be significantly changed before it comes around to you.


  You can substitute 3 cards of any one color for a locomotive when building. (stolen from the ferries in the other versions)


This should have no negative effect, though I don't know that it will actually speed things up, since yes, you'll finish that route, but the train car to played card ratio will actually go down. More tickets will likely be completed, though.


  7 cards stay face up instead of 5


This will also favor longer routes, exacerbating that problem. It will increase decision time a bit, but speed up route completion for everyone equally. You'll need to decide on a rule for how many locomotives have to be showing to wipe, likely 4 instead of 3. 

Just to be clear, I'm not trying to knock all of your ideas. I just see rippling balance issues that are worth considering. If you implement any of them, I suggest getting a few more games under your belt first so you can more clearly see the consequences, if any.