Are there any differences in karma? Is there a difference in karma of giving someone $ 1 and giving someone $ 1000, or its same for anything given?

and

Is there a difference in karma of killing a small animal (like mosquito or small fish) and killing an big animal (like a cow or an elephant), or the karma is same for all killings ?

If there is a difference what makes the difference?
The difference will always be the mental state of the person performing the action. Giving is dependent on one's state of mind while giving; the amount is intrinsically inconsequential (though obviously a poor person will feel more strongly about giving $1000 than giving $1). As Sakka, king of the devas of Tavatimsa says in the Vimanavatthu:

&quot;Natthi citte pasannamhi, appikā nāma dakkhiṇā;
Tathāgate vā sambuddhe, atha vā tassa sāvake&quot;ti. (vi. va. 804);
There is not, when the mind is confident, such a thing as a small gift,
In regards to a Tathagata, fully enlightened Buddha or, furthermore, his disciples.

Giving to the Buddha is considered to be of greater merit, not because of the benefit to the Buddha, but because of the ability to inspire faith and confidence in the giver.
As for killing, the mental state involved in killing large animals is generally more intense; also the sense of the weight of the act is greater, so the mind will be more strongly affected by it. Killing virtuous people, parents, etc. is even worse, because it sets one squarely against virtue.
But there is no hard and fast rule here either; if you obsess over killing a single mosquito buzzing over you at night, you may cultivate more unwholesomeness than a single act of killing a larger animal.
Furthermore, karma isn't a single act of killing; it may be cultivated every moment; every thought concerning a moral or immoral act is karmically potent, so it's imprecise to ask if x or y is good or bad, better or worse, since karma is only a single thought moment at a time, and thus the potency depends upon the nature of the individual mind states.