I have been induced for all five for goung to 42+ wks.
As it's your third it may not take long but third babies can be tricky and mine fitted tgat rule. But I just had my waters broken as was dilated enough and didn't need the drip except with number one and number five (she was back to back).
You can still use the pool if labour starts with just the gel /pessary or with your waters being broken I did and no 4 was born in birth pool. With two and three I laboured in water but wanted to get out before delivery,just had urge to walk about.
Even if you need continuous monitoring it does not mean flat on back on bed, I was knelt up leant over end of bed or sat in birth ball etc, yes it made it trickier with wires and keeping track of heartrate but it can be done. Tbh if babies heart rate isn't good flat on your back is worst position for baby.
Even with the syntocinon drip I still had active labour's on birth ball, kneeling etc.
Yes to having gas and air for examinations or waters being broken, they will only perform an arm on delivery floor so you should have gas and air if you want. It xan be uncomfortable sp use it if needed. They use an amnio hook, it looks more like a crochet hook than a knitting needle and it's normally disposable plastic so it's sterile. Sometimes waters can be tough to break, my waters have been as I have tough membranes apparently but sometimes it's easy and it's not always painful, it's like a good sweep but ask for gas and air if you want.
They will want to moniter you after arm or the pessary to make sure baby is happy, normally for 29-30 mins, again you don't have to be flat on your back.
Once pessary is in and monitoring done or arm is done get up and get mobile if you can to help encourage labour. I walked up and down the flights of stairs at my hospital! Or bounce on birth ball etc whatever you can/want to do.
If you don't want to be induced you can ask for expectant management for a few days, so monitoring and scan to check baby is Ok and give it a few more days.
After your waters have been broken diff hospitals have different policies on how long they will leave you for labour to establish before they want you on the drip. If you are fine and baby is fine ie no temp etc you can bargain with them over this for extra time, I did with ds3 and then contractions got going on their own and he arrived within three hours.
Remember that should be getting informed consent, it doesn't sound like they have spoken to you about pros and cons etc of induction and gone through your options. don't be afraid to ask for more information and to stand up for yourself esp re staying mobile etc.
Good luck op, despite being induced all my labour's have been a good experience, I was bolshy and insisted on staying mobile etc but that and lots of gas and air got me through.
Oh what position is baby in? Try and use optimal fetal positioning/spinning babies info to get baby to be in good position ie not bsck to bsck that will help.
Good luck.