strategy question

Is it better to go for value in 3-4 way pot when one is all in or try to eliminate a player? When we have the nuts or close to it , we are ofcorse going to try to get a few more chips from remaining players in pot , but what if we have top pair or medium or small pair.
thanks in advance for your suggestions

That really depends on the tournament stage, stack sizes, ranges of your opponents and what the flop texture is.

betting out on the flop is fine if you think you are ahead, you just have to realise that your opponents will only call with a legitimate hand or draw. So you are unlikely to get value from most weaker (lets knock out the short stack type) hands, and you will have to tighten the range you put them on, as they have to beat two hands at showdown rather than 1.

If the all in pot is negligible compared to the effective stack size of remaining players in the hand, then by all means play the hand aggressively as you normally would, just be aware that their checking may be implicit collusion rather than weakness.

lowering the chance of the all in player being eliminated by forcing other hands to fold, has ICM considerations near the bubble or in ITM

In my somewhat limited experience i find that players that have the intention of checking down will abandon that plan once one bet postflop has been made. Its like you break the truce to eliminate the other player by betting into a dry sidepot. So don't bet marginal hands.

And obviously don't bet drawing hands yourself as you would otherwise be seeing free cards, and missing would triple/quad up the short stack fi everyone else folded to your bets.

Feel free to point out any problems in my thinking

Hi , thanks on your suggestions I think your thinking on this is fine and I agrre with you that if you think your ahead you should bet for value, espacialy if you have top pair and there are flush or straight draws on the flop or turn, why should someone out draw us for free.
lowering the chance of the all in player being eliminated by forcing other hands to fold, has ICM considerations near the bubble or in ITM (please explain)

Pretty much what I was trying to say if forcing a drawing hand to fold, that would have eliminated the all in player isn't ideal in some cases.
For example if you are on the bubble, knocking a short stack out places you into the money, which is preferable than to increase your odds of winning the pot.
Or if you are top 5 at a tourney final table, and the pay out increases are significant, having the maximum chance of eliminating the short stack is the better strategy to increase your payout

When someone is ALL IN, in most cases, the remainder of the table goes to sleep just trying to knock out. If you have value, bet to win it all,,,,, Most players will not call a larger bet if their side pot investment is small

OK here's my take on this.
1. If you are in the money and you think you have flopped best hand-by all means bet to protect your flopped hand. More than likely others will fold (dry side pot) unless they have hit the flop with strong hand.
2. Close to the money bubble-Here your trying to eliminate players to get to the money. Checking it down is best thing to do even risking your flopped hand. Only way I bet at this point is if I'm extremely shortstacked myself and need to protect. But, keep in mind, other 2 to act players more than likely have you covered and will call your bets down if they have hit or have favorable odds.

I've seen many a flopped hand lose with someone all in by betting and person you bet out would have taken out the all in. I try to play money first, then aggression.

I don't think this is optimal, As the bubble is where we should be more aggressive and be trying to build our stack to attack that final table. And when the main pot is a significant portion of their stack many players find it hard to fold their 2nd best hand.
If you have a big stack/are playing aggressively the bubble shouldn't worry you.