Build a cmu block wall on two sides, face it with stone and do a concrete counter top acid stained and sealed and cantilevered 12" so you can pull bar stools up to it

You can't really "cover" the gas heater - but you could try to "disguise" it using a 'low growing' bamboo or informal hedging plant in a container that you can keep pruned and - that way you can move the plant if you need to access the heater for maintenance. If you were to put a storage unit under it you could easily disguise the heater with rosemary, or grass plants etc. Be sure to check what load bearing capacity your balcony has - so that you can safely use your balcony to its full advantage. Mirrors make a difference on the opposite wall to where your heater is - you could even use a one on that wall too - if you use a mirror opposite a garden wall your space will instantly feel bigger. To keep the maximum floor space you can on the balcony, think about a vertical wall garden and planter boxes on the railings. If you do decide to go for planters on the floor of your balcony remember to go light weight plastic as terracotta, and porcelain can get heavy quickly. For your soil - think about core peat with a slow release fertilizer - core peat is light weight but has no nutrients for your plants. I cant see that you've got a tap - if you don't then think carefully about drought tolerant plants as carrying out watering pots several times a day in summer isn't much fun. If you do have a tap - think about installing a drip irrigation system on a timer to give yourself more time to sit back and enjoy your balcony rather than working on it.

your house has a Georgian feel and suggest you check formal garden styles for inspiration and the local nursery for specific plants to suit your climate and aspect...big ceramic troughs could look good but in ground plantings are more likely to survive with less attention once established

depends on direction of sun and local climate...where is north and where is the house?..i'd buy a collection of big pots (plastic or steel garbage bins can look good) and add some rubble for drainage then deep mulch and sit your plants on top of the mulch and arrange the planters in groups to suit your furniture arrangements under the pergola and consider adding shade cloth screens around the whole pergola to shade hot sun, block wind, exclude bugs and make the space more comfortable for yourself and the plants...looks like a great house in an interesting location, congratulation