You can't kneel on the green to see how solid it is. Rule 3: If kneeling on the green leaves causes dents in the surface, fix the dents before going on to the next hole. Aside from that, you can kneel on the green. It's even allowed to putt while kneeling, but the USGA advises against it. They say it interferes with your sense of distance and angle, which could affect your stance and grip.
The main reason people don't kneel on the green is because they might get stuck. If the ground isn't firm enough, you could get stuck and hurt yourself.
But there's no rule that says you have to kneel on the green. You can stand up while putting if you want to. The only people who should be kneeling are those with physical limitations such as short legs or arthritis. Otherwise, you're free to use whatever position allows you to putt the best shot.
You are not permitted to kneel on the green to mark your partner's line of putt. People have been known to figure out how to get the ball into the hole by kneeling on the green and telling their partner, "putt toward my knee." That is not possible. The penalty for doing so is a loss of putt.
The rule is there to prevent people from getting an unfair advantage by watching where their partner putst. If you want to know where to place your own shot, then watch what they do. But if they're going to putt where you want them to, then why should you be penalized because you watched them putt?
In addition, people tend to stand more closely aligned with their partners' shots when they can see what direction they're going to putt. This may influence the way they approach the putt themselves. So by kneeling you deny your partner information that could help them putt better.
There is no rule that prevents you from kneeling before making a short putt, although most players will say they'd rather not take any chance of missing.
However, if you're going to make multiple putts in a row (or if it's raining or windy), you should get up off the green to avoid getting wet/cold.
Overall, yes you can kneel while putting.
No Deliberate Greens Testing Exception-Testing Greens Between Two Holes: A player may rub the surface of the hole's putting green or roll a ball on any practice green (see Rule 5.5b).
In general, you can't do anything with the green except play golf from the fairway. However, there are times when it is necessary to reach into the hole for various reasons. As long as you don't fill in any part of the hole, reaching into the hole to retrieve something that has fallen into it is permitted. For example, if a dog eats a golf ball while you're playing and drops it inside the hole, you can reach into the hole to get it.
Here are some other examples of things that might be done with the hole during your round:
Putting - You can use the edge of your putter to push dirt out of the way if there is enough room. This is generally only necessary for larger greens.
Footing - You can use your foot to block out portions of the hole if space allows. For example, if there is a large branch in the way of an open shot, you could lift your foot to make sure you have enough room to shoot the ball.