Probability question on cards.?

This applies to Poker (Texas Holdem) where two cards are dealt to each players (7 players to be considered). Next the 5 Community cards (common for everyone) are dealt out, 3 first (Flop), then after another round of betting 1 card is dealt (Turn), again after another round the remaining 5th card is dealt (River). Now if one gets two unconnected cards of the same suit but with a rank difference less than 5, what is the probability that the remaining cards may yield a Three of a Kind, Fours, Flush or a Straight or even a Full House? There are four different situations, 1. After the Deal, 2. After the Flop, 3. After Turn, 4. After River.

The situation changes dramatically after the flop as you may turn up with 3 or 4 cards neccesary but will expect that to turn up during turn or river.

However you need to give me hints for calculation for situation 1 only. I am arriving at very low probabilities as I am multiplying the probability factors for each subsequent cards. I also want to know the combined probability of any one of the above happening and again combined including Two Pairs.

I don't think you can add up the individual probabilities as there are many overlapping situations.

Detailed reasonings or are not required and thanking you and wishing luck in a hand of poker if this interests you into actually playing the game. I've learned it recently and am enjoying it.