Seems to me you have two areas where you want to optimise; I've run large games before (8 players in a Rolemaster campaign, I must have been mad!) and keeping all the players involved is tough, here's what I learnt from it.

Combat

• Preparation; I don't know about AD&D 4e but I do know about complicated systems and preparation can save you loads of time. Make a carddeck of monster information you can quickly refer to, no referencing is quicker play.
• Numbering. Number everything and order them and write a short name after it for fluff. Get a bit of paper and write down the initiative order for everyone in a fight and list your monsters as numbers; then track resources next to their name. While a player is deciding you can decide what the next monster is doing.
• Tracking: Monsters can be very complicated to track with multiple resources, hit points, abilities and so on. So keep them simple, put check boxes in your initiative list to mark what they've used so you don't have to keep checking; if it helps. Also you can do the same for the players; make a table where you track each players hits, ac and so on so you don't have to keep asking if you've hit, if they're at 50% hits or whatever.
• Mooks and minions. It's been mentioned before and it's a great timesaver; Splat the 2 hit point Kobold really has only two states, annoying and dead. Don't bother counting.
• Simplify, simplify, simplify: Round monsters stats off, ignore time consuming abilities, if they're nearly dead make them really dead. If a monster is unconscious don't bother with stabilisation rolls (for example) fudge things away dramatically to save time, keeping the fight flowing is more important.
• Rounding: If maths isn't your forte round damage to the nearest five or ten either way and it'll save you a lot of headaches and counting; you can even then use a simple bar/gate system to add on damage in blocks of five to count damage.
• Poke players in order, tell the player who's next after whoever's deciding to get ready each time. "Bob you're up, Eric start thinking of what you want to do."
• If you really want to harsh it, give players a decision time frame; 1 minute per turn, maybe less. This will encourage them to prep their decisions!

Social

I notice in your question you have When we do story based adventure there are 2-3 players that do most of the talking and the others twiddle their thumbs. To me, social situations are where a GM can be the most lazy, especially with a big group.

• Encourage PC to PC interaction, backstories are gold for this; players should be talking to each other and deciding stuff as much as possible.
• Draw out the quiet ones, it's been said before but "victimise" those who keep quiet, encourage them, drop bits from background into their story "Bob you notice an old book on the shelf that is titled 'A history of Bobland' where your old village Bobford was" and so on.
• Round Robin it. Break the group into smaller ones and give each one a bit of time; Two groups are out buying stuff? Tell the first about the shop, do a bit of interaction then switch to the other group for a bit - this can take a bit of getting used to but I've found it very helpful in stopping players sitting around being bored - just switch every few sentences or so between groups, the other group gets a chance to think some more and the previously waiting group gets a go.
• By the same dint give NPCs in social situations more people. How intimidating is it for 8 people to try and talk to the same person at the same time? Add in the Wizards familiar to talk to, another shop assistant, add in extra NPC's to situations so you can use the Round Robin tactic.

Well that sounds about par for the course. I have a regular group that I DM for since 2009, and your player archetypes sound pretty familiar to me. (And yes, those are the real archetypes, not what is described in the DMG).

Don't worry so much

First off, don't worry so much. From your post, it seems like it's going ok. The important part is that you have fun. If the group has fun, there's not a big problem, even if you as a DM might feel you are doing stuff wrong.

Talk about it

After a session, or maybe during a coffee or smoke break, ask the players about these topics. What do they like, what do they dislike? How do they think the game can be improved? If everyone says it's fine, great. If not, I am sure they have some suggestions for improvement. I do this after every session, sometimes for 3 minutes, sometimes we discuss this for half an hour or more. It really helps.

Play to the player's strengths

If a player or his character is good at something (and everyone has has a speciality), give him or her a chance to use this strength. So the rogue player is inventive - give him a chance to use his wits. What I do is that I generally prepare some specific additional puzzles or challenges for them. I think the best way to integrate those is if you have a fight and at the same time some additional pressing matter that people have to attend to, either with some skill checks or just with good ideas. So maybe while the party deals with the brunt of the attackers, someone must slip through the enemy ranks and disable the ballista before the city walls are down.

Help your players

So the ranger treats this like a video game, but is not pro-active enough. Well, he doesn't have to go shopping. Drop some loot for him that is a part (one third) of a magic bow. Maybe a bow string made out of dryad hair? This should awaken his video gamey instinct to collect all parts, involve him more into the story and finally lead to him getting a better bow.

Some answers on this board stress how important player agency and pro-active players are for the game. I am not sure I fully agree with this. Some people are just naturally more passive, and if you can help them so they have more fun and everyone has more fun, just do it.

Use mechanics to cover the player's weaknesses

For the wizard - maybe propose Sense Motive, Gather Information or similar (Insight in 5e) checks when you feel he's missing something. If he succeeds the check, give him a hint. Like this, you can use the character's strength to cover the player's weaknesses. And after a few successes, it is likely that the positive reinforcement will lead to the player doing this on his own more often.

A similar thing about this assertiveness: Have a friendly NPC cast a spell him, or maybe an enemy a curse, and tell him how this makes him feel more assertive, mighty and filled to the brim with power. Sometimes this helps such players to slip into a role they would not otherwise take on.

On the other hand, specifically this is not really a problem in my eyes - a nerdy shy wizard doesn't seem so out of place.

Offer help with their characters

So this is a two-sided sword. It might be a great help, but can also ruin a lot. Make sure the players want and appreciate your help with their characters before doing anything. You really want to avoid messing with 'their guy' if they don't want you to. But if done right, it can be a great help to certain players.

One of my players is like your barbarian player. Doesn't like to read the rules, but is very enthusiastic and a great addition to the game. Before a new campaign, I generally have a skype conf with him and discuss the background a bit and what character he wants to play. Then I build a character for him, send it to him and we discuss some changes he usually wants. Once we have a character he's happy with, I make him a nice colourful character sheet in Numbers with all the important stuff up front. This worked out very well. It allows him to focus on the game, and not worry about details or picking whatever CharOp stuff from way too many sources he's not interested in, while still playing his concept and being on a similar power level as other players who are more into optimising.

Allow them to rebuild or change characters (within reason)

Sometimes, a new player might choose a character that doesn't really fit his play style. Might be the wizard player would be happier with a sorcerer - new players tend to fare better with the simpler approach of having a few known spells instead of building a spellbook and preparing their daily prepared lists.

The system

Finally, maybe D&D 3.5e is not the ideal system for players who don't read up on the rules themselves. If your group is open to change the system, I can suggest two options:

1. D&D 5e. While similar in many respects, it has one advantage: A lot of stuff is just way simpler and easier than 3.5. While 3.5e is my personal favourite among all D&D editions, for my main group, I have switched to 5e, and we haven't looked back.

2. Games with far less rules: Dungeon World or Fate come to mind. I have played them, and had fun, but for my group, it was not the right choice. We like the miniature battles and everything around that. But those are certainly great games.

And, in the end, the evil master plan

Of course, all this cuddling and making players happy only serves one goal: to make them experienced enough so you can start being a true evil dungeon master. Once they know the system, know their characters and have a fair chance to survive bad things, the fun starts: Poison, traps and and horrible dismemberment. It's no fun punishing them if they have no chance to deal with it. But once they have shown they are up for the task, you can start driving the finger screws in, and let them into the tomb of horrors.