In general, yes. If interest rates go higher, then any existing fixed-rate bonds - and hence ETFs holding those bonds - become less valuable. The further each bond is from maturity, the larger the impact. As you suggest, once the bonds do mature, the fund can replace them at a market price, so the effect tails off.
The bond market has a concept known as "duration" that helps reason about this effect. Roughly, it measures the average time from now to each payout of the bond, weighted by the payout. The longer the duration, the more the price will change for a given change in interest rates.
The concept is just an approximation, and there are various slightly different ways of calculating it; but very roughly the price of a bond will reduce by a percentage equal to the duration times the increase in interest rates. So a bond with a duration of 5 years will lose 5% of its value for a 1% rise in interest rates (and of course vice-versa).
For your second question, it really depends on what you're trying to achieve by diversifying - this might be best as a different question that gives more detail, as it's not very related to your first question. Short-term bonds are less risky. But both will lose value if the underlying company is in trouble. Gilts (government bonds) are less risky than corporate bonds.