1♦. A nice and straightforward question: open in your longest suit; 1♦. There is no doubt that after a 1♠ response you are left with having to make the rather unsatisfactory rebid of 2♦, but that is the best you can do (you cannot rebid 2♥ because that is above your first suit and would be a reverse). 

Some players seem to think that it is all right to open 1♥ on hands like this, but that can never be right because you will be implying five cards in the heart suit when you rebid 2♦. Open 1♦ and you will finish in 2♦, which is certainly a better contract than 2♥.

East Hand ♠ A8765 ♥ J83 ♦ J62 ♣ 97

4♠. Your partner has opened with a pre- empt showing 5-9 HCP and a seven-card spade suit and you have a weak hand with little support, so what is there to think about? You can be pretty sure that your opponents should be in 4♥ – your partner should not have four hearts, so the opponents have at least an 8-card fit and they will have the strength and distribution to make 10 tricks likely. At the favourable vulnerability, they would score 600+ for a game contract while your non-vulnerable undertricks will be cheap against that price.
If you play in 4♠ doubled you will need to make only 7 tricks to make a profit: three off doubled costs 500 points which would be a handsome profit against their game – a good sacrifice.
Raise to 4♠ and make your opponents’ life difficult – they may not even feel able to double you. As you can see, you should be able to make a comfortable 7 tricks, whilst your opponents would have been able to make 10 or 11 tricks in hearts.

East Hand ♠ AK87652 ♥ 92 ♦ 87 ♣ 42

2NT. South opens with a pre-emptive style opening at the two-level. It is  important to get used to dealing with weak twos: they are becoming much more popular. The most sensible defence against weak twos is to play double for take-out and have a natural 2NT overcall showing a strong balanced hand (about 16-19 HCP). Your overcalls should generally be based around a six-card or longer suit.
With 17 HCP you need to do something and you have a choice of two calls: 2NT or 3♣.
With your spade stop being the ace, you may well to be able to control the play and make the most of your excellent club suit in a no-trump contract. When given a choice of two reasonable bids, it generally makes sense to choose the one that has the better upside. Bidding 2NT will often result in playing in 3NT, which has a great upside, whilst 3♣ is less likely to score well, although it is certainly safer.
If your partner raises to 3NT you should be able to establish your clubs without South getting the lead, therefore making at least nine tricks.

East Hand ♠ 76 ♥ K532 ♦ AJ52 ♣ T43

3♣. Your partner has rebid in a suit higher than his first suit at the two-level: this constitutes a reverse and is a strong bid: showing 16+ HCP. You are forced to make a response. With a weak hand, you should give preference to your partner’s first suit and hope that keeps him quiet. Bid 3♣ and a good partner should probably pass although some might still want to risk 3NT. 3♣ is the best contract.
 

East Hand ♠ 43 ♥ A2 ♦ AK62 ♣ KQJ82